# Properties of Exponents and Roots

An exponent is a mathematical shorthand from indicating that a will be multiplied by itself a fixed number of times. If we were to look at the example of: 4^{3}, The larger base number (4) is the number that will be multiplied by itself 3 times, which is indicated by the exponent found in the upper right corner of it. A root is another common symbol that you will find in math. They are roots which are somewhat of the related to exponent. A square root asks you what number multiplied by itself equals the value under the square root symbol. For instance, √16. What would you multiply by itself to get 16. 4 x 4 = 16. So the square root would be 4, in this case. These worksheets and lessons will help students understand the basic properties and nature of both exponents and roots.

### Aligned Standard: HSN-RN.A.1

- Rewriting Exponents Step-by-step Lesson- We work with fractional exponents in this lesson. We show you how roots and squares relate.
- Guided Lesson - A nice relation back and forth with exponents and roots. We also work on a fractional exponents.
- Guided Lesson Explanation - We show you a repeated way for solving problems of this magnitude.
- Practice Worksheet - Find the end value of all the inverse roots and squares we provide you with. They can be quite challenging.
- Matching Worksheet - This will be fun, but difficult at the same time. See how it goes.
- Exponents Five Pack - You are given various expressions. We want you to find the values of those expressions.
- Nature of Roots Sum and Product of Roots Worksheet Five Pack - All addition and multiplication here.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Practice rewriting exponents in root and vice versa.

- Homework 1 - We currently have the value in exponent form. We want to convert it to a root value. The good thing is that exponents and roots are just the opposite of one another.
- Homework 2 - We have to rewrite in exponent form. If we remember, the root form and exponent form are the inverse of one another.
- Homework 3 - We have to first simplify the given exponent form.

### Practice Worksheets

I provided brief explanations here to help students along with the skill faster.

- Practice 1 - Rewrite the root in exponential form and calculate the value.
- Practice 2 - The problem we are working on is a bit trickier because the exponent value is a fraction.
- Practice 3 - Calculate the end value of problems.

### Math Skill Quizzes

Each quiz mixes all the skills that you will find for this level.

- Quiz 1 - We can combine the values into one exponent.
- Quiz 2 - Find the value of expression.
- Quiz 3 - Where does it end?

### Common Properties of Exponents and Roots

**Properties of exponents** - The properties of exponents are also called operations for these components. Just like the order of the operations, there is a need for you to memorize such operations in order to be successful. There are five properties of exponents:

**1. Product of Powers** - When you multiply two exponents that share the same base number. You can keep the base number and just add the exponents. Example: 4^{3} x 4^{5} = 4^{8 (3 + 5)}

**2. Quotient Rule ** -As we know the counterpart of multiplication is division. So, it stands to reason that if you ran into a similar situation, but with division you find the different of the exponents. When you are dividing one exponent by another that shares the same base number, you just find the difference between the exponents. Example: 6^{5} ÷ 6^{2} = 6^{3 (5 - 2)}

**3. Power Rule ** - If you have a base number with an exponent and it is raised to a power, you multiply the exponent and the power together. Example: (3^{2})^{4} = (3)^{2 x 4} = 3^{8}

**4. Negative Exponent Rule ** - This is mostly used for solving more advanced equations. If you have a negative exponent, you can invert the base to form a positive. Example: 3^{-6} = 1 / 3^{6}

**5. Zero Exponent Rule ** - Any non-zero value raised to a power of zero is equal to 1. Example: 8^{0} = 1.

Roots are one of the most common mathematical elements to be used and it is commonly denoted by the symbol √. If the symbol is written alone with another number inside, then it means that it is a square root. If the symbol has a number before it like: 3√ then it means that it is asking for a cube root.

Here are some of the properties of roots:

**Multiplication** - √a x √b = √(ab)

**Addition** - √a + √b ≠ √ (a + b)

**Subtraction** - √a - √b ≠ √ (a - b)

**Division** - √a/√b = √(a/b).