Exponent Rules

When multiplying expressions with the same base, add the exponents.

$x^a·x^b=x^{a+b}$

Let's look at a few examples:

• $2^3·2^4=2^{3+4}=2^7=128$
• $x^5·x^2=x^{5+2}=x^7$

Note that the reverse is also true: if the exponent is the sum of two numbers (or constants or variables), we can rewrite the expression as a product of two powers.

$x^{a+b}=x^a·x^b$

Even for students who know the product rule, they often have more trouble applying it in this way.

Let's look at an example:

$2^{x+2}=2^x·2^2=2^x·4$

The same concept applies if we are adding more than two things in the exponent:

$x^{a+b+c}=x^a·x^b·x^c$

When dividing expressions with the same base, subtract the exponents.

$\frac{x^a}{x^b}=x^{a-b}$

Let's look at a few examples:

• $\frac{3^5}{3^2}=3^{5-2}=3^3=27$
• $\frac{y^8}{y^3}=y^{8-3}=y^5$
• $\frac{2^x}{2^3}=2^{x-3}$

Again, the same concept applies in reverse: if the exponent is the difference of two numbers (or constants or variables), we can rewrite the expression as a quotient of two powers.

$x^{a-b}=\frac{x^a}{x^b}$

So, for example:

$3^{x-2}=\frac{3^x}{3^2}=\frac{3^x}{9}$

When raising an exponential expression to another power, multiply the exponents.

${\left(x^a\right)}^b=x^{a·b}$

Let's look at a few examples:

• ${\left(2^3\right)}^2=2^{3·2}=2^6=64$
• ${\left(x^4\right)}^3=x^{4·3}=x^{12}$
• ${\left(2^x\right)}^3=2^{x·3}=2^{3x}$

And, again, the same concept applies in reverse: if the exponent is the product of two numbers (or constants or variables), we can rewrite the expression as a power of a power.

$x^{a·b}={\left(x^a\right)}^b$

And, because of the commutative property of multiplication ($a·b=b·a$), it doesn't matter which exponent we raise first.

$x^{a·b}={\left(x^a\right)}^b={\left(x^b\right)}^a$

Let's look at some more examples:

• $2^{x·3}=2^{3·x}={\left(2^3\right)}^x=8^x$
• ${27}^{\frac{1}{3}·x}={\left({27}^{\frac{1}{3}}\right)}^x={\left(\sqrt[3]{27}\right)}^x=3^x$
  • This example shows how we can rewrite (27^x)^(1/3) as (27^(1/3))^x using the power rule. Since 27^(1/3) = ∛27 = 3, we get 3^x as our final answer.

When raising a product to a power, distribute the exponent to each factor.

${(x·y)}^a=x^a·y^a$

Let's look at a few examples:

• ${(2·3)}^4=2^4·3^4=16·81=1296$
• ${(x·y)}^3=x^3·y^3$

When raising a quotient to a power, distribute the exponent to both the numerator and denominator.

${\left(\frac{x}{y}\right)}^a=\frac{x^a}{y^a}$

Let's look at a few examples:

• ${\left(\frac{2}{3}\right)}^3=\frac{2^3}{3^3}=\frac{8}{27}$
• ${\left(\frac{x}{y}\right)}^4=\frac{x^4}{y^4}$

When you have an expression like $a+b$, you cannot distribute the exponent to each term.

${(a+b)}^c\ne a^c+b^c$

For example, take the expression ${(2+3)}^2$.

The proper way to simplify this expression is to first add the terms inside the parentheses, then raise the result to the power:

${(2+3)}^2=5^2=25$

If we were to distribute the exponent, we would get a different result, which is incorrect:

$2^2+3^2=4+9=13$

Since 25 ≠ 13, we can see that ${(a+b)}^c\ne a^c+b^c$

Rules that apply to addition generally also apply to subtraction, so the same rule applies to expressions like $a-b$:

${(a-b)}^c\ne a^c-b^c$

Any non-zero number raised to the power of 0 equals 1.

$x^0=1$ (where x ≠ 0)

Let's look at a few examples:

• $7^0=1$
• $1^0=1$
• ${(-3)}^0=1$
• $0^0=\mathrm{undefined}$
  • Or some times it is defined as 1, depending on the context.
  • Don't worry about this edge case on the SAT.

When you see a negative exponent, make the exponent positive and move the base and its exponent to the opposite side of the fraction.

$x^{-a}=\frac{1}{x^a}$

$\frac{1}{x^{-a}}=x^a$

Let's look at a few examples:

• $2^{-3}=\frac{1}{2^3}=\frac{1}{8}$
• $x^{-4}=\frac{1}{x^4}$
• $2x^{-3}=\frac{2}{x^3}$
• $\frac{5}{3x^{-2}}=\frac{5x^2}{3}$

A fractional exponent can be rewritten as a radical expression. The denominator of the fraction represents the root of the radical, and the numerator represents the power that the base is raised to.

$x^{\frac{a}{b}}=\sqrt[b]{x^a}$

Instead of having the exponent inside the radical like above, we can also rewrite it so that the entire radical is raised to the power of the numerator instead:

$x^{\frac{a}{b}}={\left(\sqrt[b]{x}\right)}^a$

To see why we can rewrite it either way, let's apply the power rule.

Using the power rule, we can rewrite the expression as:

$x^{\frac{a}{b}}={\left(x^{\frac{1}{b}}\right)}^a={\left(\sqrt[b]{x}\right)}^a$

Or, we can also rewrite it as:

$x^{\frac{a}{b}}={\left(x^a\right)}^{\frac{1}{b}}=\sqrt[b]{x^a}$

Let's look at a few examples:

• $8^{\frac{2}{3}}={\left(\sqrt[3]{8}\right)}^2=2^2=4$
• ${16}^{\frac{1}{2}}=\sqrt{16}=4$
• ${\left({27}^x\right)}^{\frac{1}{3}}={\left({27}^{\frac{1}{3}}\right)}^x={\left(\sqrt[3]{27}\right)}^x=3^x$
  • This example uses a combination of the power rule and the fractional exponent rule to simplify the expression.