## Properties of Exponents

#### Aligned To Common Core Standard:

**High School** - HSA-SSE.A.3c

What Are the Properties of Exponents?
Have you ever noticed a '^' sign on computers and a calculator?
It is not a misrepresentation of the alphabet 'V.' This inverted V represents an operation of math. Besides addition, subtraction, division, and multiplication there is one more mathematical operation known as exponentiation.
72 means 7 to the power 2, or to simply put: 72= 7 X 7 =49. 7 is the base, and 2 is its exponent.
Exponents show the number of times to use the base factor. Let's dig deeper into the properties of exponents
**FIVE PROPERTIES OF EXPONENTS** - To get a clearer idea about exponents, we need to dissect the five properties of exponents. Let us look at them one by one
PRODUCT OF POWERS - Here is the formula for the product of powers: (x^{a})(x^{b}) = x^{(a + b)}
To put it in words, when you multiply exponentials with the same base, their exponents add up. For example, if you have (x^{2})(x ^{3}), you will add up the exponents of 'x'. that is we get x5, which is same as adding 3+2
**POWER TO POWER** - Here is the formula for power to power: (x ^{a})^{b}
When you get a power to power, you multiply the exponents. For example, if you have (x ^{4})62, it would mean to multiply x^{4} two times. The result we will get is x ^{(2*4)} = x^{8}
**QUOTIENT OF POWERS** - Here is the formula for quotient of powers: (x^{a})/(x^{b})= x^{(a-b)}
Lets see how this works. When you have (x^{4})/(x^{3}), it simply means to subtract the exponent of numerator from the exponent of denominator. That is, the result you will get is x^{(4-3)} = X^{1}
**POWER OF A PRODUCT** - To find the power of a product we use the formula: (ab)m = am .bm
For example, if you have (3x)^{2}, then you split the base and multiply it the number of times of exponents. That is, (3x)^{2} = 32. X 2 = 9x2
**POWER OF A QUOTIENT** - Power of a quotient is similar to power of product. Suppose you are dividing two expressions: 203/43 = 5.4 * 5.4 * 5.4/4.4.4. cancelling out all the 4s we get, 53

### Printable Worksheets And Lessons

- Exponent Products
Step-by-step Lesson- Learn to simplify products of exponents;
includes negative exponents.

- Guided Lesson
- We go over the operations of quotients and products with exponents.

- Guided Lesson
Explanation - I show you the standard format for multiplication
and division with exponents. Then we work that into the actual problem.

- Practice Worksheet
- Find five products and five quotients of exponents.

- Laws of Rational Exponents
Five Pack - These are more advanced. They include negative fractional
exponents.

- Matching Worksheet
- Match each problem to the final outcome of its operations.

#### Homework Sheets

The division of exponents really confuses kids when they first see it.

- Homework 1 - Simplify. Express your answer using a positive exponent.
- Homework 2 - Divide the numerator by the denominator.
- Homework 3 - Multiply the r's, remembering to add the exponents.

#### Practice Worksheets

The best way to get kids to remember the operations is rote memorization, I find any way.

- Practice 1 - To multiply powers with the same base, add their exponents. A negative exponent can be written as a positive exponent in the denominator.
- Practice 2 - Simplify. Express your answers using positive exponents.
- Practice 3 - Is that a Jack in the Box?

#### Math Skill Quizzes

The quizzes really help the skills to develop further.

### How is This Skill Used in The Real World?

This skill lends itself to the financial, scientific, and technology communities. Being able to model exponential growth is a critical skill for all these industries. In the financial industry this is created when attempting to understand compounding interest. If calculations are off, it can make or break a bank. Scientists model population size and the future of it through this technique as well. Those projections have significant ramification on the state and local governments in those areas. This can also be reversed to look at the potential for exponential decay that can show the inverse of the two examples we have discussed. Decay is used to model the movement of light and sound wave trajectories for communication purposes in the communication industries. Most recently this technique has been used to model the growth and decline of coronavirus. Government leaders made closing and reopening decisions almost entirely based on how the exponential data was tracking to determine the significance of personal behavior changes. The curve was also used to demonstrate that social distancing and facemasks were effective towards reducing the spread of the disease.