Roots (Radicals)

I. Intro

A radical expression is any mathematical expression containing a radical symbol (√).
Radicals are simply the inverse operation to exponents. More technically, the root of a number A is another number x, which when multiplied by itself n number of times, equals A.

Hence,  where 

There are three parts of a radical expression. The radical symbol, which means "root of." The radicand, which is the number under the radical symbol, is the number you are finding the "root of." The index, which is the small number written as part of the radical symbol, indicates the number of times the unknown number is multiplied by itself to equal the radicand. If no index is shown, it is assumed to be 2, or square root.

Roots
The 2nd root is generally called the square root.
The 3rd root is called the cube root.
Roots higher than three are referred by ordinary numbers such as "fourth root", "fifth root", etc.

Roots can also be written in exponential form, such as:

II. Square Roots
When we are trying to find the square root of a number (say, 25), we are trying to find a number that when multiplied by itself gives that original number. In the case of 25, we find that 5⋅5=25, so 5 is the square root of 25.

The square root is the inverse of the square (exponent of 2), much like multiplication is the inverse of division. A perfect square is any whole number that has been squared. Consequently, all perfect squares have square roots that are whole numbers.

So $\sqrt{25}=5$. 

Here are some other simple examples:

Principal Root

The square root of a number is the value that, when multiplied by itself, gives the original number. For example, both +5 and −5 are square roots of 25 because:

5×5=25

(−5)×(−5)=25


Thus, every positive real number n has two square roots—a positive square root and a negative square root. The unique positive square root is called the principal square root or principal root.

The radical sign (√) generally represents the principal square root. Unless specified otherwise, “the square root” of a number refers exclusively to the principal square root.

Connection To A Square

Square Root of a Decimal

Khan: Square Root of Decimal

Square Root Decimal by Converting to Fraction

Square Root of a Fraction

Onlinemath: Simplify Square Roots That Have Fractions

III. Cube Root

When we are trying to find the cube root of a number (say, 27), we are trying to find a number that when multiplied by itself three times gives that original number. In the case of 27, we find that 3⋅3⋅3=27, so 3 is the cube root of 27.

Factoring to find Cube Root

If we can't figure out what factor multiplied by itself three times will result in the given number, we can make a factor tree.

Example:
  

So the prime factorization of 64 is 2 x 2 x 2 x 2 x 2 x 2

We are looking for he cube root of 64, so we want to split the prime factors into three identical groups. 

Notice that we can rearrange the factors like so:

Connection To A Cube

Finding the cube root of 27 is the same as finding the edge of a cube with a volume of 27.

IV. Practice

Khan: 8th Grade Math: Square Root and Cube Root

V. Reference 
Simplifying a Square Root by Factoring
Purplemath: Square Roots: Introduction & Simplification
Math Open Reference: Root
Math Open Reference: Radical
Wikipedia: nth root algorithm
Study.com: Simplifying Radical Expressions with Variables
For Dummies: How to Convert Square Roots to Exponents
Study.com: Radical Expression: Definition & Examples
Khan: Cube Root Review
   

Inequalities: Solving Multi-Step Inequalities

I. Solving Two-Step Inequalities

Solving multi-step inequalities is the same as solving multi-step equations except for the one important rule previously discussed in the section on solving one-step inequalities. That is, whenever you multiply or divide by a negative number, you must flip the inequality sign to so that the statement remains true.

Lumen: Solve Inequalities

CK-12: Solving Two-Step Inequalities

CK-12: Solving One and Two-Step Inequalities

Equations: Solving Multi-Step Linear Equations

I. Two-Step Equations

A two-step equation is an algebraic equation with two operations in it that requires two steps to solve. As discussed in the intro to equations section, the goal is to isolate the variable on one side of the equation.

To isolate the variable, we have to undo the involved operations by using their inverse operations (opposite operations) and solve for the variable. In general, we want to perform addition or subtraction first, then perform multiplication or division. 

So the basic steps to solve two-steps equations are to balance both sides of the equation using following rules: 

• Undo the addition by subtracting both sides with the same number.
• Undo subtraction by adding both sides with the same number.
• Undo the multiplication by dividing both sides with the same number.
• Undo the division by multiplying the same number to both sides.

Example: Solve 2x + 1 = 5

Step 1: Subtract 1 from both sides.

2x + 1 - 1 = 5 - 1

2x = 4

Step 2: Divide both sides by 2

2x/2 = 4/2

x = 2

Example: Solve 3x/4 - 2/3 = 7/3

Step 1: Add 2/3 to both sides

3x/4 - 2/3 + 2/3 = 7/3 + 2/3

3x/4 = 9/3

Step 2: Multiply 3/4 to both sides

(4/3) 3x/4 = 9/3 (4/3)

x = 4

Example: 

II. Solving Multi-Step Linear Equations

The multi-step equations are algebraic equations that require multiple steps to solve. Solving multi-step equations in algebra is similar to solving one-step and two-step equations, but the process is a little lengthy as there are multiple steps involved.

A. Equations With Variables On Both Sides

Some equations may have the variable on both sides of the equal sign, as in this equation:

4x − 6 = 2x + 10  

To solve this equation, we need to “move” one of the variable terms. This can make it difficult to decide which side to work with. It doesn’t matter which term gets moved, 4x or 2x, however, to avoid negative coefficients, you can move the smaller term.

B. Equations With Fractions
Sometimes, you will encounter a multi-step equation with fractions. If you prefer not working with fractions, you can use the multiplication property of equality to multiply both sides of the equation by a common denominator of all of the fractions in the equation (Least Common Denominator). This will clear all the fractions out of the equation.

1. Find the Least Common Denominator (LCD): Identify all the denominators in the equation. The LCD is the smallest multiple that all the denominators divide into evenly.

2. Multiply Every Term by the LCD: Multiply both sides of the equation (and every individual term within them) by the LCD. This will cancel out all the denominators.

3. Solve the Resulting Equation: After clearing the fractions, you’ll have an equation with whole numbers. Solve this equation using the standard steps (combine like terms, isolate the variable).

4. Check Your Solution: Substitute your answer back into the original equation (with fractions) to verify that it is correct.

Example with variable in denominator

C. Equations With Decimals

Sometimes, you will encounter a multi-step equation with decimals. If you prefer not working with decimals, you can use the multiplication property of equality to multiply both sides of the equation by a a factor of 10 that will help clear the decimals. See the example below.

D. Classify Solutions to Linear Equations

There are three cases that can come up as we are solving linear equations. We have already seen one, where an equation has one solution. Sometimes we come across equations that don’t have any solutions, and even some that have an infinite number of solutions. The case where an equation has no solution is illustrated in the next examples.

No Solution

This is not a solution! You did not find a value for x. Solving for x the way you know how, you arrive at the false statement 4=54=5. Surely 4 cannot be equal to 5!

Infinite Number of Solutions

Example

Solve: 5x + 3 -4x = 3 + x

Combine like terms

x + 3 = 3 + x

Subtract 3 from both sides

x + 3 - 3 = 3 - 3 + x

x = x

E. Equation Word Problems

Splashlearn: Two-step equations

Ck-12: Solving Two-Step Equations

Lumen: Multi-Step Linear Equations

Splashlearn: Solving Multi-Step Equations

Onlinemath:Solving Equations With Fractions

Openstax: College Algebra

CK-12: Solving Rational Equations Using the Least Common Denominator

Coordinate Plane

Before learning about two variable equations it's important to have a basic understanding of the coordinate plane.

The coordinate plane was developed centuries ago (in 1637, to be exact) and refined by the French mathematician René Descartes. In his honor, the system is sometimes called the Cartesian coordinate system. The coordinate plane can be used to plot points and graph lines. This system allows us to describe algebraic relationships in a visual sense, and also helps us create and interpret algebraic concepts.

The Components of the Coordinate Plane

A coordinate plane is a two-dimensional plane formed by the intersection of two number lines. One of these number lines is a horizontal number line called the x-axis and the other number line is a vertical number line called the y-axis. The point at which the two axes intersect is called the origin. The origin is at 0 on the x-axis and 0 on the y-axis.

Locations on the coordinate plane are described as ordered pairs. An ordered pair tells you the location of a point by relating the point’s location along the x-axis (the first value of the ordered pair) and along the y-axis (the second value of the ordered pair).

In an ordered pair, such as (x, y), the first value is called the x-coordinate and the second value is the y-coordinate. Note that the x-coordinate is listed before the y-coordinate. Since the origin has an x-coordinate of 0 and a y-coordinate of 0, its ordered pair is written (0, 0).

Example:

Find the point (4,3)

To identify the location of this point, start at the origin (0, 0) and move right along the x-axis until you are at 4. The 4 indicates that, from the origin, you have traveled four units to the right along the x-axis. This is the x-coordinate, the first number in the ordered pair.

From 4 on the x-axis move up 3 units as indicated by the y-coordinate. With an x-coordinate of 4 and a y-coordinate of 3, you have the ordered pair (4, 3).

Identifying Quadrants

The intersecting x- and y-axes of the coordinate plane divide it into four sections. These four sections are called quadrants. Quadrants are named using the Roman numerals I, II, III, and IV beginning with the top right quadrant and moving counter clockwise.

Independent vs Dependent Variable

The names of the axes vary based on your application. The horizontal number line represents your independent variable and the vertical number line represents your dependent variable. The independent variable is the variable that causes a result, that is measured by the dependent variable.

Example 1

Identify the independent and dependent variables in the situations below.

a. The revenue, or amount of money a business earns from selling a product, versus the number of products sold

b. The number of hours versus the number of bacteria in a petri dish

Solution:
a. It is usually easier to identify the dependent variable. First, ask whether the number of products sold depends on the revenue OR whether the revenue depends on the number of products sold. Since the revenue depends on the number of products sold, the revenue is the dependent variable and the number of products sold is the independent variable.

b. Does the number of hours depend on the number of bacteria or does the number of bacteria depend on the number of hours? Since the number of bacteria depends on the elapsed time, the number of bacteria is the dependent variable and the number of hours is the independent variable.

Finding The Slope Of A Line

See Two Variable Equations

Reference

Splashlearn: Coordinate Plane

Lumen: Beginning Algebra: Coordinate Plane

CK-12: The Coordinate Plane

Finding the Unit Rate

A unit rate is a way of comparing two different quantities where one of the quantities (the denominator) is expressed as one unit. The unit rate expresses how much of one quantity exists per unit of the other quantity. 

To find the unit rate you:

1. Identify the quantities

2. Write the quantities as a fractional division problem with the numerator being the quantity you are trying to calculate per unit. The denominator is the total number of units of the second quantity (the one you want to unitize).

3. Complete the division. 

Example1 :

Jayda takes 3 hours to deliver 189 newspapers on her paper route. What is the rate per hour at which she delivers the newspaper?

The quantities are 3 hours and 189 newspapers. The question asks for the "rate per hour" so we know hours will become the unit quantity (the denominator) and number of newspapers is what we are trying to calculate (the numerator). 

189/3 = 63 

Therefore Jayda can deliver 63 newspapers per hour

Example 2

Nierria earns $75 for 4 hours of tutoring. How much does Nierria earn for 1 hour of tutoring?

We are looking for how much she makes "per hour" so we know hours is the unit quantity (denominator) and the amount earned per hour is what we are trying to calculate (the numerator).

75 dollars/4 hours = 18.75 dollars/per hour

Example 3

You walk 1/2 mile in 1/4 hour. How many miles will you walk in one hour?

We are looking for miles per hour so hours will be the denominator and miles the numerator.

1/2 / 1/4 = 1/2 x 4/1 = 4/2 = 2 miles per hour

Example 4

Calvin cleans 3/5 of his bathroom with 1/3 of a bottle of cleaning solution. At this rate, what fraction of the bottle of cleaning solution will Calvin use to clean his entire bathroom?

We are trying to figure out how much or many bottles (numerator) to clean his one bathroom (denominator).

1/3 bottle / 3/5 bathroom = 1/3 bottle x 5/3 bathroom = 5/9 bottle to clean his bathroom.

https://www.khanacademy.org/math/pre-algebra/pre-algebra-ratios-rates/pre-algebra-rates/v/finding-unit-rates

https://www.wikihow.com/Calculate-Unit-Rate

https://www.splashlearn.com/math-vocabulary/unit-rate-formula

https://www.mathmammoth.com/videos/ratio/unit_rate_fractions

Area Model (Box) Method Multiplication

Box method of multiplication:
Step 1: Make a table with the place values of the first factor along the top and the place values of the second factor down the left.
Step 2: Multiply the place values and enter their products in the table.
Step 3: Sum each row of the table.
Step 4: Add these row sums. This is the product of the original factors.

Example:

72 x 19

Khan: Multiply with area model: 16 x 27





Reference

IXL: Box Method Multiplication

https://www.bbc.co.uk/bitesize/articles/zmm9dxs#zcn77yc

Units of Measurement

Measurement is defined as the system or act of measuring. It can be understood as a process of defining physical items using numbers.

The process of measuring consists of three main steps. First, you need to select an attribute of something that you wish to measure (for example, the area of a playground). Second, you need an appropriate unit of measurement (in this case, square yards). Finally, you need to determine the number of units. This last step is usuaully accomplished by using some sort of measuring tool. 

A. Measurement Attributes

Measurable attributes refer to characteristics of objects that can be measured, such as length, area, volume, mass, and temperature. When comparing two objects with a measurable attribute in common, we see which object has “more of” or “less of” the attribute.

Length - The distance between two points

Area - The amount of surface something has.

Volume - The space occupied within the boundaries of an object in three-dimensional space. It is also known as the capacity of the object.

Mass - A measure of how much matter is in an object
Temperature - a measurement of how hot or cold something is.

B. Unit of Measurement

There are two main "Systems of Measurement" used in the United States: Metric System and English/US Customary Standard.

1. Metric System

The metric system was developed by the French Academy of Sciences in 1791. A single, universally accepted system of measurement was needed because commerce among countries had increased substantially. Interestingly, Thomas Jefferson tried to convince the United States Congress at that time to adopt the base ten system of measurement, but his proposal was rejected. Today the United States is the only major nation that does not use the metric system extensively.

Changes have occurred in the metric system over time because of advances in technology. What has evolved is the International System of Units, which is abbreviated as SI. The units in the SI system are based on scientific formulas and natural constants. Strictly speaking there are differences between the metric system and the SI system. However, because there are so many similarities and the differences are slight from a nonscientific point of view, the SI system is often called the “metric system.”

The metric system defines standard units for length, area, volume, mass and temperature. Those units are meter (length), square meter (area), cubic meter (volume), gram (mass) and degrees Celsius (temperature). Larger and smaller units are obtained by multiplying or dividing the standard units by powers of ten. The ease with which conversions can be made (by moving a decimal point) within this measurement system are one of the reasons it has been adopted so universally. Greek prefixes indicate units larger than the standard unit—multiples of 10 (deka-), 100 (hecto-), and 1,000 (kilo-). Latin prefixes indicate units smaller than the standard unit—0.1 (deci-), 0.01 (centi-), and 0.001 (milli-). These relationships are summarized below:

Note: There are many more prefixes. See Wikipedia: Metric Prefixes

a. Metric Length (Meter)

The standard unit of length in the metric system is the meter (m). Applying the prefixes mentioned earlier we have: 

Kilometer (km) - 1,000 meters

Hectometer (hm) - 100 meters

Decameter (dam) - 10 meters

Meter (m) - base unit

Decimeter (dm) - 1/10 meter

Centimeter (cm) - 1/100 meter

Millimeter (mm) - 1/1,000 meter

Of these, the most commonly used are the kilometer, meter, centimeter and millimeter. 

-A millimeter is about the thickness of a credit card. 

-A centimeter is the width of the nail of my little finger. 

-A meter is about the distance from the fingertips of my outstretched arm to my shoulder. 

b. Metric Area (Square Meter)

The standard metric unit for area is the square meter (m2). This unit is a square that is one meter in length on each side. Applying the prefixes we have:

Square kilometer (km²) - 1,000,000 square meters

Square hectometer (hm²) - 10,000 square meters

Square decameter (dam²) - 100 square meters

Square meter (m²) - base unit

Square decimeter (dm²) - 0.01 square meter

Square centimeter (cm²) - 0.0001 square meter

Square millimeter (mm²) - 0.000001 square meter

c. Metric Volume (Cubic Meter)

The volume of objects can be found using liquid measures such as liters and solid measures such as cubic meters. Often when measuring liquids or describing the size of containers that hold liquids, we refer to volume as capacity. However, people use both terms, readily switching between them.

The standard metric unit for volume is the cubic meter—a cube that is one meter on each size. This unit is used to measure the volume of very large objects. Imagine filling an “open top” cubic decimeter with water. The amount it holds (or its capacity) is a liter (L). There is no international standard regarding when to use liters and when to use cubic meters (or cubic decimeters, etc.). In practice, liters are most commonly used for items measured by the size of their container (such as fluids and berries), whereas cubic meters are most commonly used for items measured by their dimensions.

Cubic kilometer (km³) - 1,000,000,000 cubic meters - 1 trillion liters

Cubic hectometer (hm³) - 1,000,000 cubic meters - 1 billion liters

Cubic decameter (dam³) - 1,000 cubic meters - 1,000,000 liters

Cubic meter (m³) - base unit - 1,000 liters

Cubic decimeter (dm³) - 0.001 cubic meters - 1 liter

Cubic centimeter (cm³) - 0.000001 cubic meters - 1 milliliter

Cubic millimeter (mm³) - 0.000000001 cubic meters - .000001 liters

-A standard bottle of water (16 Fl Oz) is a little less than half a liter. 

d. Metric Mass & Weight (Gram)

The terms weight and mass are often incorrectly used interchangeably. The distinction is that weight is the measure of the gravitational force exerted on an object whereas mass is the amount of material in an object. More technically, mass is the property of an object that resists acceleration. To measure mass we use a balance and compare the object to be measured with a standard unit. To measure weight we use a
spring scale and observe how much pull is exerted on the object. A 120-pound astronaut has the same mass when on Earth or when on the moon, but her weight on the moon is one sixth of her weight on Earth because of the difference in gravity (that is, the astronaut’s mass remains the same regardless of her location whereas her weight changes). 

The base unit of mass in the metric system is the gram (g). One kilogram is equivalent to 1,000 grams. A kilogram is equal to the mass of one cubic decimeter (also known as one liter) of water at 4° Celsius (when it is most dense).

kilogram (kg) = 1,000 g 
hectogram (hg) = 100 g

dekagram (dag) = 10 g

gram (g) = base unit
decigram (dg) = 0.1 g
centigram (cg) = 0.01 g
milligram (mg) = 0.001 g

-A small paper clip weighs about 1 gram. 

-A pineapple weighs roughly 1 kilogram

e. Metric Temperature 
Celsius scale was Invented in 1742 by Swedish astronomer Anders Celsius and hence named after him. Celsius, also called centigrade, is based on the freezing point of water which is 0°, and the boiling point of water which is 100° The temperature in Celsius is represented with °C. Some sources seem to imply that it is part of the metric system and other say that only the Kelvin is the only official unit of temperature in the SI system.

2. The English/US Customary Standard System

United States customary units form a system of measurement units commonly used in the United States and most U.S. territories since being standardized and adopted in 1832. The United States customary system developed from English units that were in use in the British Empire before the U.S. became an independent country. The United Kingdom's system of measures evolved by 1824 to create the imperial system (with imperial units), which was officially adopted in 1826, changing the definitions of some of its units. Consequently, while many U.S. units are essentially similar to their imperial counterparts, there are noticeable differences between the systems.

Units of length are inches, feet, yards and miles. Units of area include the square inch, square foot, square yard, acre, and the square mile. Units of volume in the English system are the cubic inch, cubic foot, cubic yard, and cubic mile. More common units of volume, especially for liquids, include
the teaspoon, tablespoon, fluid ounce, cup, pint, quart, and gallon. The units of mass/weight in the English system are the ounce and the pound (Note: there's a difference between the common avoirdupois ounce and the troy ounce which is used to measure precious metals). Temperature in the English system is measured using degrees Fahrenheit.

a. US Customary Length

Inch (in)

Foot (ft) = 12 inches

Yard (yd) = 3 feet

Mile (mi) = 1760 yards

-An inch is about the distance between the first and second joint of my little finger when I bend it. 

b. US Customary Area

Square inch (in²)

Square feet (ft²) = 144 square inches

Square yard (yd²) = 9 square feet

Square acre = 4840 square yards

Square mile (mi²) = 640 acres = 5,280 feet

c. US Customary Volume

Cubic inch (in³)

Cubic foot (ft³) = 1,728 cubic inches

Cubic yard (yd³) = 27 cubic feet
Cubic Mile (mi³) = 5,451,776,000 cubic yards

Teaspoon (tsp)

Tablespoon (Tbsp) = 3 teaspoons

Fluid ounce (Fl Oz) = 2 tablespoons

Cup  = 8 fluid ounces

Pint (pt) = 2 cups = 16 fluid ounces

quart (qt) = 2 pints

Gallon (gal) = 4 quarts

d. US Customary Temperature

The Fahrenheit scale is a temperature scale based on one proposed in 1724 by the European physicist Daniel Gabriel Fahrenheit (1686–1736).It uses the degree Fahrenheit (symbol: °F) as the unit. Historically, on the Fahrenheit scale the freezing point of water was 32 °F, and the boiling point was 212 °F (at standard atmospheric pressure). This put the boiling and freezing points of water 180 degrees apart.

https://www.mathsisfun.com/measure/unit.html

https://www.ck12.org/section/units-of-measurement/

https://www.onlinesafetytrainer.com/a-comprehensive-guide-to-units-of-measurement/

https://www.cuemath.com/measurement/

https://www.utm.edu/offices-and-services/stem-center-for-teaching-and-learning/_media/Measurable_Attributes_notes.pdf

https://www.mathsisfun.com/measure/index.html

https://homeschool.ie/wp-content/uploads/2020/08/Lesson-13-final-revised-version.pdf

http://www.vendian.org/mncharity/dir3/bodyruler/

https://en.wikipedia.org/wiki/Square_metre

https://www.cuemath.com/temperature-conversion-formulas/

https://www.splashlearn.com/math-vocabulary/measurements/customary-units

chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://www.nist.gov/system/files/documents/2019/11/05/appc-20-hb44_final.pdf

https://home.howstuffworks.com/what-is-size-order-of-socket-wrenches.htm

https://www.math.net/us-customary-units

https://en.wikipedia.org/wiki/Fahrenheit

Math is fun: Metric Area

Math Matters

Inequalities: Intro & Solving One-Step Inequalities

I. Intro

A relationship between two expressions or values that are not equal to each other is called inequality. It is used most often to compare two numbers by their size

The symbols used in inequalities are:

> Greater than

< Less than

≥ Greater than or equal to

≤ Less than or equal to

≠ Not equal to

Writing Solution Set In Interval Notation

While writing the solution of an inequality in the interval notation, we have to keep the following things in mind. 

•If the endpoint is included (i.e., in case of ≤ or ≥) use the closed brackets '[' or ']'
•If the endpoint is not included (i.e., in case of < or >), use the open brackets '(' or ')'
•Use always open bracket at either ∞ or -∞.

Inequalities On A Number Line

While graphing inequalities, we have to keep the following things in mind.

•If the endpoint is included (i.e., in case of ≤ or ≥) use a closed circle.
•If the endpoint is NOT included (i.e., in case of < or >), use an open circle.

•Draw a line from the endpoint that extends to the right side if the variable is greater than the number.
•Draw a line from the endpoint that extends to the left side if the variable is lesser than the number.

The inequality 𝑥>−1 would be read, “𝑥 is greater than negative one.” Notice there are many values that would make this statement true. For example, 𝑥=2 or 𝑥=5 or 𝑥=10. In fact, any number to the right of -1 on the number line would be a solution to this inequality. We can describe the solution set as an interval of real numbers. In this case, the interval (−1,∞)

II. Solving One-Step Inequalities

Solving linear inequalities is similar to solving linear equations. In fact, all the properties that we used to solve linear equations also hold for linear inequalities.

Notice these properties are the same as the properties of equality with one exception: If you multiply or divide both sides of an inequality by a negative number, then you need to switch the direction of the inequality. Why is this true? Let's consider what happens with numbers. We can agree 3 < 5. However, if we multiply both sides of the inequality by -1, we get

−1 × 3 < − 1 × 5

−3 ≮ −5

The same inequality would result if we divided both sides by -1. So, when we multiply or divide both sides of an inequality by a negative number, we need to switch the direction of the inequality.

Solving linear inequalities works the exact same way as solving linear equations with this one exception.

Splashlearn: Inequality

Cuemath: Inequalities

Ck12: Solving One- and Two-Step Inequalities

Mathwarehouse: Graph Inequalities on Number Line

Lumen: Solve Inequalities