Free video lesson

How to Apply Exponent Rules on the SAT

Exponents look messy, but five rules collapse almost any expression — and radicals are just fractional exponents in disguise. Learn the system once and the ugly ones fold up in seconds.

Math · Equivalent expressions2:10Published July 6, 2026

On YouTube: SAT Exponents: Watch the Base

What this lesson covers

  • Multiply same base → add; divide → subtract; power of a power → multiply
  • Negative exponent = reciprocal
  • Radicals: nth root of x^a = x^(a/n) (bottom = root, top = power)
  • Worked: (x³)²·x⁻¹/x² = x³ ; 27^(2/3) = 9
  • Practice: 16^(3/4)

Questions worked in the video

  1. 0:29Simplify (x³)² · x⁻¹ / x².
  2. 0:49Evaluate 27^(2/3).
  3. 1:09Evaluate 16^(3/4).

Worked examples

The questions the video works, written out: the setup, each step, the answer and the trap.

0:29Example 1

Simplify (x3)2⋅x−1x2\displaystyle \frac{(x^3)^2 \cdot x^{-1}}{x^2}.

  1. Start with the power of a power. (x3)2=x3⋅2=x6(x^3)^2 = x^{3 \cdot 2} = x^6. A power of a power multiplies the exponents, it does not add them.
  2. Now every factor has the same base, x, so the exponents can be combined. Multiplying same bases adds the exponents: x6⋅x−1=x6+(−1)=x5x^6 \cdot x^{-1} = x^{6 + (-1)} = x^5.
  3. Dividing same bases subtracts the exponents: x5x2=x5−2=x3\displaystyle \frac{x^5}{x^2} = x^{5 - 2} = x^3. In one line, the final exponent is 6−1−2=36 - 1 - 2 = 3.
  4. Check with x = 2: (23)2=64(2^3)^2 = 64, times 2−1=12\displaystyle 2^{-1} = \frac{1}{2} gives 32, divided by 22=42^2 = 4 gives 8, and 23=82^3 = 8.

Answer: x3x^3

Every piece shares the base x, so the rules apply directly: multiply the exponents for the power of a power, add for the product, subtract for the quotient. The common slip is treating (x3)2(x^3)^2 as x5x^5 by adding instead of multiplying, which leaves x2x^2 at the end. The other slip is reading x−1x^{-1} as a negative quantity; it is a reciprocal, 1x\displaystyle \frac{1}{x}, which is exactly why its exponent gets subtracted.

0:49Example 2

Evaluate 272/327^{2/3}.

  1. A fractional exponent is a radical in disguise. The bottom of the fraction is the root and the top is the power: xa/n=(xn)a\displaystyle x^{a/n} = (\sqrt[n]{x})^a.
  2. So 272/3=(273)2\displaystyle 27^{2/3} = (\sqrt[3]{27})^2. Take the root first because it shrinks the number: the cube root of 27 is 3, since 3⋅3⋅3=273 \cdot 3 \cdot 3 = 27.
  3. Now apply the power: 32=93^2 = 9.
  4. Check by reversing the exponent: 93/2=(9)3=33=27\displaystyle 9^{3/2} = (\sqrt{9})^3 = 3^3 = 27, so 9 raised to the reciprocal exponent gives 27 back.

Answer: 9

Since the denominator 3 is the root and the numerator 2 is the power, 27 to the two thirds is the cube root of 27, squared, which is 9. The trap is swapping the roles and taking the square root of 27 cubed, which is not a whole number, or multiplying 27 by two thirds to get 18. Root from the bottom, power from the top.

1:09Example 3

Evaluate 163/416^{3/4}.

  1. Bottom is the root, top is the power: 163/4=(164)3\displaystyle 16^{3/4} = (\sqrt[4]{16})^3.
  2. Fourth root of 16: what number multiplied by itself four times gives 16? 2⋅2⋅2⋅2=162 \cdot 2 \cdot 2 \cdot 2 = 16, so it is 2.
  3. Cube it: 23=82^3 = 8.
  4. Check: 84/3=(83)4=24=16\displaystyle 8^{4/3} = (\sqrt[3]{8})^4 = 2^4 = 16.

Answer: 8

The fourth root of 16 is 2 and 2 cubed is 8. Students who cube first get 163=409616^3 = 4096 and then need its fourth root, which is still 8 but is far harder to see under time pressure; take the root first when the base is a perfect power. The wrong answer 12 comes from multiplying 16 by three fourths, treating the exponent as a fraction of the base.

Chapters

Lesson transcript

The narration of the video, word for word, under its chapter headings.

0:00Five rules

Welcome to SAT Climb. Exponents look messy, but five rules collapse almost anything. Learn them once, and the ugliest expression folds up in seconds.

0:16The setup: an ugly expression

Simplify this: (x³)² · x⁻¹ / x². Looks awful. Watch it disappear.

0:29Collapse it

Five rules. Multiply same bases, add the exponents. Divide, subtract. A power of a power, multiply. So (x³)² is x⁶. Then 6 − 1 − 2 = 3. The whole thing is just x³.

0:49Radicals = fractional exponents

Radicals are secretly fractional exponents. The bottom is the root, the top is the power. So 27^(2/3) is the cube root of 27, squared. That's 3² = 9.

1:09Your turn

Your turn. 16^(3/4). 4th root first, then cube it. What do you get?

1:35Three traps

Three traps. Adding exponents with different bases — that only works when they match. A power of a power multiplies, it doesn't add. And a negative exponent is a reciprocal, not a negative number.

1:56Recap

Exponents: solved. Five rules, any expression. Start free at satclimb.com.

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