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# Properties of Exponents

Exponents have a lot of properties that define how they can be manipulated in mathematics, and remembering them all can feel overwhelming. This page is structured as a convenient reference point to help students remember how to approach math problems involving exponents.

## Product of powers property

The product of powers property states that expressions involving exponents can be simplified by adding the exponents together, provided that all of the exponents are modifying the same base value. Expressed another way, $\left(2x-3\right)\left(2x+1\right)$

Consider ${7}^{2}×{7}^{5}$ . This expression means $\left(7×7\right)×\left(7×7×7×7×7\right)$ , or the product of seven 7s once we eliminate the parentheses. Therefore, we can use ${7}^{7}$ to express the same thing more easily.

## Power to a power property

The power to a power property states that expressions involving exponents raised to another power can be simplified by multiplying the two exponents, provided that all of the exponents are modifying the same base value. This works because ${\left({3}^{2}\right)}^{4}$ is the same thing as ${3}^{2}×{3}^{2}×{3}^{2}×{3}^{2}$ , giving us a total of eight 3s to multiply. Therefore, we can simply say ${3}^{8}$ .

## Quotient of powers property

The quotient of powers property states that expressions involving exponents can be divided by subtracting one exponent from the other provided that they're modifying the same base value. For example, let's say that you're dividing ${5}^{4}$ and ${5}^{3}$ . Written out, you're dividing $\left(5×5×5×5\right)$ by $\left(5×5×5\right)$ . Three of those 5s cancel each other out since 5 divided by 5 equals one, leaving us with just one $5$ (or ${5}^{1}$ ) left.

## Zero exponents

Many students assume that anything with an exponent of zero should be equal to zero because anything multiplied by zero is zero. However, that's not the way non-zero numbers with exponents work. Using the quotient of powers property, we can see that $\frac{{a}^{m}}{{a}^{m}}={a}^{\left(m-m\right)}={a}^{0}$ . Since any nonzero number divided by itself is equal to 1, it follows that ${a}^{0}=1$ for any nonzero number a.

That said, zero exponents can be tricky and often depend on context. However, in most cases, any non-zero number raised to the power of zero is equal to 1.

## Properties of exponents practice questions

a. What is ${2}^{2}×{3}^{2}$ ?

${2}^{2}×{3}^{2}={2}^{\left(2+3\right)}={2}^{5}=32$

b. What is ${\left({4}^{3}\right)}^{2}$ ?

${\left({4}^{3}\right)}^{2}={4}^{\left(3×2\right)}={4}^{6}=4096$

c. What is ${8}^{7}$ divided by ${8}^{6}$ ?

$\frac{{8}^{7}}{{8}^{6}}={8}^{\left(7-6\right)}={8}^{1}=8$

d. What is ${127}^{0}$ ?

$1$

## Get help with the properties of exponents

Many students feel intimidated by exponents, potentially hurting their self-confidence in other areas of mathematics. A 1-on-1 tutor could help the student in your life better understand what the properties of exponents are and when and how to apply them, increasing their self-confidence for the academic challenges ahead. Reach out to the knowledgeable Educational Directors at Varsity Tutors to learn more about private instruction and what it could do for your student.

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