Simplifying Exponents

For rules of exponents applied to algebraic functions instead of numerical examples, read Rules of Exponents - Algebraic.

The laws of exponents are rules that can be applied to combine and simplify expressions with exponents. These rules are true if $a$ is positive, and $m$ and $n$ are real numbers.

$$\def\arraystretch{1.5} \begin{array} { c | c } \text{ Rule name} & \text{ Rule } \\ \hline \text{Product Rule} & a^m \times a^n = a^{ m + n } \\ & a ^n \times b^n = (a \times b)^ n \\ \hline \text{Quotient Rule} & a^n / a^m = a^ { n - m } \\ & a^n / b^n = (a/b) ^ n \\ \hline \text{Negative Exponent} & a^ {-n} = \frac{1}{a^n} \\ \hline \text{Power Rule} & (a^n)^m = a^ { n \times m } \\ \hline \text{Tower Rule}& a ^ { n^ m } = a ^ { \left ( n^ m \right) } \\ \hline \text{Fraction Rule} & a ^ { 1/n} = \sqrt[n]{a } \\ & \sqrt[m]{ a^n} = a^ { n/m} \\ \hline \text{Zero Rule} & a^0 = 1 \\ & 0^ a = 0 \text{ for } a > 0 \\ \hline \text{One Rule} & a^1 = a \\ & 1^a = 1 \\ \end{array}$$

When the bases of two numbers in multiplication are the same, their exponents are added and the base remains the same. If $a$ is a positive real number and $m,n$ are any real numbers, then

$$\large a^m \times a^n = a^{ m + n } .$$

What is $2^3 \times 2^4?$

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We have

$$2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2^7.$$

In other words,

$$2^3 \times 2^4 = 2 ^{3+4} = 2^7. \ _\square$$

What is $3^2 \times 3^6?$

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We have

$$3^2 \times 3^6 = 3 ^{2+6} = 3^8. \ _\square$$

When the exponents of two numbers in multiplication are the same, then bases are multiplied and the exponent remains the same. If $a, b$ are positive real numbers and $n$ is any real number, then we have

$$\large a ^n \times b^n = (a \times b)^ n.$$

Here are some examples based on the above rule.

What is $(2 \times 3)^5?$

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We have

$$(2 \times 3)^5 = 2^5 \times 3^5 = 2^5 3^5. \ _\square$$

What is $2^3 \times 5^3?$

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We have

$$2 ^ 3 \times 5 ^ 3 = ( 2 \times 5 )^3 = 10 ^3 = 1000. \ _\square$$

Here is a problem for you to try:

When the bases of two numbers in division are the same, then exponents are subtracted and the base remains the same. If is $a$ positive real number and $m,n$ are any real numbers, then we have

$$\large \dfrac{a^n}{a^m} = a^ { n - m }.$$

Go through the following examples to understand this rule.

What is $2^5 \div 2^3?$

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We have

$$2^5 \div 2^3 = \frac{2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2} = 2 \times 2 = 2^2.$$

In other words,

$$2^5 \div 2^3 = 2^{5-3} = 2^2. \ _\square$$

Simplify $\dfrac{ 3^4 }{ 3 ^2} .$

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We have

$$\frac{ 3^4 }{ 3 ^2} = 3 ^{4-2} = 3^2 =9. \ _\square$$

Practice your mind at the following problems.

When the exponents of two numbers in division are the same, then the bases are divided and the exponent remains the same. If $a, b$ are positive real numbers and $n$ is any real number, then we have

$$\large \frac{a^n}{b^n} = \left(\frac ab\right) ^ n.$$

Here are the examples to demonstrate the above.

What is $\left( \dfrac{3}{5} \right)^2?$

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We have

$$\left( \frac{3}{5} \right)^2 = \frac{3^2}{5^2} = \frac{9}{25}. \ _\square$$

What is $\left( \dfrac{5^3}{7^2} \right)^4?$

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We have

$$\left( \frac{5^3}{7^2} \right)^4 = \frac{5^{3 \times 4}}{7^{2 \times 4}} = \frac{5^{12}}{7^8}. \ _\square$$

Let's try the following simple problem:

For any nonzero real number $a,$ a negative exponent is handled like so: $a^{-n} = \dfrac{1}{a^n}.$ Here, the fraction $\dfrac{1}{a^n}$ is called the reciprocal of $a^n:$

$$\large a ^ { -n } = \frac{ 1} { a^n }.$$

In written words, we say "$a$ to the negative $n$ equals the reciprocal of $a$ to the $n.$"

What is $2^{-3}?$

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We have

$$2^{-3} = \frac{1}{2^3} = \frac{1}{8}. \ _\square$$

What is $\dfrac{1}{{10}^{-1}}?$

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We have

$$\begin{aligned} \dfrac{1}{{10}^{-1}}& = \dfrac{1}{\hspace{2mm} \frac{1}{10}\hspace{2mm} }\\\\ & = 10.\ _\square \end{aligned}$$

What is $3^{5} \div 3^{7}?$

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We have

$$3^{5} \div 3^{7} = 3^{5-7} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9}. \ _\square$$

Simplify $\dfrac{ 4^{-3} }{ 16^{-1}}.$

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We have

$$\left( \frac{ 4^{-3} }{ 16^{-1}} \right) = \frac{ 4^{-3} }{ \left(4^2\right)^{-1}} = \frac{ 4^{-3} }{ 4^{-2}} = 4^{-3 - (-2)} = 4^{-1} = \frac{1}{4}. \ _\square$$

Power rule of exponents is stated as

$$\large (a^n)^m = a^ { n \times m }.$$

Caution!

For the power rule, with $n = 2$ and $m = \frac{1}{2}$, the LHS is $\big(a^2\big) ^ { \frac{1}{2} } = | a |$, while the RHS is $a^ { 2 \times \frac{1}{2} } = a$. These are not equal. There are also special cases to consider when dealing with negative or complex values.

Now, these are the examples based on the above rule:

What is $\big(2^3\big)^4?$

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We have

$$\big(2^3\big)^4 = 2^3 \times 2^3 \times 2^3 \times 2^3 = 2^{3+3+3+3} = 2^{12} .$$

In other words,

$$\big(2^3\big)^4 = 2^{3 \times 4} = 2^{12}. \ _\square$$

What is $2^5 \times 4^3?$

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We have

$$\begin{aligned} 2^5 \times 4^3 &= 2^5 \times \big(2^2\big)^3 \\ &= 2^5 \times 2^6 \\ &= 2^{11}. \ _\square \end{aligned}$$

What is $\big(2^3\big)^2 \times \big(2^2\big)^5?$

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We have

$$\big(2^3\big)^2 \times \big(2^2\big)^5 = 2^{3 \times 2} \times 2^{2 \times 5} = 2^6 \times 2^{10} = 2^{16} . \ _\square$$

Here is a problem to try:

In a tower of exponents, we work from the top down. So a tower of exponents is evaluated by

$$\large a ^ { n^ m } = a ^ { ( n^ m ) },$$

an ordering that follows somewhat naturally from the order of operations on exponents.

Towers of exponents problems lend themselves to a common misconception due to an order of operations error. In general, $a ^ {\large( b^ c ) } = \left ( a ^ b \right) ^ c$ is false. Rare cases wherein the statement is true occur when $c = 1$ or $a = 1$. One non-trivial example is when $a = b =c = 2$. In this case, we have

$$\begin{aligned} \text{LHS}&: &2 ^ { ( 2 ^ 2 ) } &= 2 ^ 4 = 16 \\ \text{RHS}&: &\big( 2 ^ 2 \big) ^ 2 &= 4 ^ 2 = 16. \end{aligned}$$

Go through the following examples to understand this rule:

What is $4 ^ { 3 ^ 2 }?$

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We have

$$4 ^ { 3 ^ 2 } = 4 ^ 9 = 262144 . \ _\square$$

Note: It is not equal to $\big( 4 ^ 3 \big) ^ 2 = 64 ^ 2 = 4096$.

Rule of exponents for fractions works in two steps as

$$\large \begin{array}{c}&a ^ {\frac 1n} = \sqrt[n]{a }, &a^ {\frac mn} = \sqrt[n]{ a^m} \end{array}.$$

Raising to a fractional exponent is similar to taking a root. The second rule follows by raising the first rule to the $m^\text{th}$ power.

What is $81 ^ \frac12?$

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We have

$$81 ^ \frac12 = \sqrt{81} = 9 . \ _\square$$

What is $343 ^ \frac {2}{3}?$

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We have

$$343 ^ { \frac{2}{3} } = \sqrt[3] { 343 ^ 2 } = \sqrt[3] { \big( 7 ^ 3\big) ^ 2 } = \sqrt[3]{ 7 ^ 6 } = 7 ^ { \frac{ 6}{3} } = 7 ^ 2 . \ _\square$$

It's interesting to note that 1 raised to any power is equal to 1. That is, for any real number $a,$ it is always true that

$$\large 1 ^ a = 1.$$

Furthermore, any non-zero number raised to the zero power is 1:

$$\large a^0 = 1.$$

Note: For $0^0,$ see what is $0^0$.

What is $1 ^ {2345}?$

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We have

$$1 ^ { 2345} = 1. \ _\square$$

Note: We do not need to multiply it out $2345$ times, to know that the product is still $1.$

What is $3^0?$

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We have

$$3^0 = 1. \ _ \square$$

What is $3^{6} \div 3^{4} - 1?$

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Using our knowledge of the order of operations, we divide first, and then subtract:

$$3^{6} \div 3^{4} - 1 = \frac{3^6}{3^4} - 1 = 3^{6-4} - 1 = 3^2 - 1 = 9 -1 = 8. \ _\square$$

To learn more about exponents, check out these related pages:

• Rules of Exponents - Algebraic
• Exponential Functions
• Rational Exponents
• Order of Operations - Exponents