Applying Exponent Rules
A handful of exponent rules cover most of what the SAT asks — the errors come from mixing up when to add and when to multiply.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
Same base, multiplying: add exponents. Same base, dividing: subtract. Power of a power: multiply. These only apply when the bases match.
How to solve one, step by step
Example: simplify $\frac{x^5 \cdot x^2}{x^3}$.
- Multiply on top (add exponents): $x^{5+2} = x^7$.
- Divide (subtract): $x^{7-3} = x^4$.
The mistakes that cost points
- Adding exponents with different bases. The rules require matching bases.
- Multiplying when you should add. Multiplying like bases adds exponents; raising a power multiplies them.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
Which of the following is equivalent to $\frac{(a^3)^4}{a^5}$?
- A$a^{60}$
- B$a^2$
- C$a^7$
- D$a^{17}$
Show solution
Answer: C, $a^7$. First apply the power rule to the numerator: $(a^3)^4 = a^{3 \cdot 4} = a^{12}$. Then apply the quotient rule: $\frac{a^{12}}{a^5} = a^{12-5} = a^7$.
Medium
Simplify $\dfrac{12(m^{6})^{3}}{3m}$
- A$4m^{8}$
- B$9m^{17}$
- C$4m^{18}$
- D$4m^{17}$
Show solution
Answer: D, $4m^{17}$. Apply the power rule: $(m^{6})^{3} = m^{18}$. The expression becomes $\dfrac{12m^{18}}{3m}$. Divide the coefficients: $\dfrac{12}{3} = 4$. Apply the quotient rule: $\dfrac{m^{18}}{m} = m^{18-1} = m^{17}$. The simplified expression is $4m^{17}$.
Hard
Simplify $\dfrac{30(-x^{5})^{3}}{6x}$
- A$-24x^{14}$
- B$-5x^{14}$
- C$-5x^{15}$
- D$5x^{14}$
Show solution
Answer: B, $-5x^{14}$. Apply the power rule, tracking the sign carefully. $(-x^{5})^{3} = (-1)^{3} \cdot x^{15}$. Since the exponent 3 is odd, $(-x^{5})^{3} = -x^{15}$. The expression becomes $\dfrac{-30x^{15}}{6x}$. Divide the coefficients: $\dfrac{30}{6} = 5$. Apply the quotient rule: $\dfrac{x^{15}}{x} = x^{15-1} = x^{14}$. The simplified expression is $-5x^{14}$.
Find your exact gaps
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