Exponents

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SAT exponent questions usually test your ability to apply the rules of exponents correctly.

1. Basic Rules of Exponents

Product Rule: $a^m \times a^n = a^{m+n}$
Quotient Rule: $\dfrac{a^m}{a^n} = a^{m-n}$ (a ≠ 0)
Power Rule: $(a^m)^n$ = $a^mn$
Power of a Product: $(ab)^m = a^m \times b^m$
Power of a Quotient: $(\dfrac{a}{b})^m = \dfrac{a^m}{b^m}$
Zero Exponent: $a^o = 1$ (a ≠ o)
Negative Exponent: $a^{-n} = \dfrac{1}{a^n}$
Fractional Exponent: $a^{\dfrac{m}{n}} = \sqrt[n]{a^m}$

Example:

$16^{\dfrac{3}{4}}=\sqrt[4]{16^3}=8$

2. Solving Exponent Equations

If bases are the same, set exponents equal.

Example:

Solve:

$2^{3x+1}=2^7$ ,

Drop the bases

$3x+1=7$ ,

$x=2$

3. Scientific Notation

Write large/small numbers as: $a \times 10^n$ (1 ≤ a < 10)

Example:

$54,000 = 5.4 \times 10^4$ ,

and

$0.0032 = 3.2 \times 10 ^{-3}$

4. Growth/Decay with Exponents

Often connects with exponential growth/decay problems: $y = a (1 \pm r)^t$