If 2^a = 8 and 2^b = 32, what is a + b?

C. 8. Since 8 = 2^3 and 32 = 2^5, a = 3 and b = 5, so a + b = 8.

SAT Math skill page

SAT Exponent Rules Practice

Use exponent rules to simplify expressions, compare powers, and avoid common sign traps.

Practice exponent rules in Dolphin SAT Practice questions

10-16 min practice time3 questions in practice blockAdvanced Math

Practice time10-16 min

Practice questions3 questions

Best forAdvanced Math

What this tests

What to know for this SAT skill

Apply product and quotient rules for exponents
Simplify powers raised to powers
Interpret negative and zero exponents
Connect radicals to fractional exponents

How do you solve SAT questions with roots and fractional exponents?

Rewrite roots as fractional powers, then choose the exponent rule that matches the operation. Add exponents when multiplying like bases, subtract them when dividing like bases, and multiply them when raising a power to another power. Finally, solve for the quantity the question actually asks for.

How do you find an unknown exponent after simplifying roots?

For what value of x is the given expression equivalent to

(70 n)^{30 x}

, where

n > 1

5 70 n (70 n)^{2}

$\frac{1}{150}$
$\frac{1}{30}$
$\frac{1}{25}$
$\frac{6}{5}$

Answer: C. $\frac{1}{25}$

Rewrite each root as a fractional power: $5 70 n = (70 n)^{1/5}$ and $70 n = (70 n)^{1/2}$ . The root index becomes the denominator of the exponent.
Apply the power rule to the squared square root: $((70 n)^{1/2})^{2} = (70 n)^{1}$ . Then multiply powers with the same base by adding exponents: $(70 n)^{1/5} (70 n)^{1} = (70 n)^{6/5}$ .
Match the exponents: $30 x = \frac{6}{5}$ . Divide by 30 to get $x = \frac{6}{150} = \frac{1}{25}$ , so choose C. Substituting this value gives $30 (1/25) = 6/5$ , the exponent just derived.

Common trap

The simplified exponent is $6/5$ , but the question asks for x, not 30x. Choice D stops one step too early. Multiply exponents for a power raised to a power; add them only when multiplying expressions with the same base.

Why do the conditions matter?

The condition $n > 1$ makes the base $70 n$ positive and greater than 1. Equal powers of that base therefore have equal exponents. That conclusion would not follow for a base of 1.

Dolphin question bank #16264. Worked explanation reviewed October 8, 2026.

How do you simplify a quotient with different root indices?

\frac{x ^{5}}{3 x ^{4}} = x^{\frac{a}{b}}

for all positive values of x, what is the value of

\frac{a}{b}

$\frac{7}{6}$
$\frac{1}{3}$
$\frac{4}{3}$
$\frac{3}{2}$

Answer: A. $\frac{7}{6}$

Express both radicals as powers of x: $x^{5} = x^{5/2}$ and $3 x^{4} = x^{4/3}$ . Keep the square-root and cube-root indices distinct.
Use the quotient rule for the same nonzero base: $\frac{x ^{5/2}}{x ^{4/3}} = x^{5/2 - 4/3}$ . Dividing the powers means subtracting their exponents, not dividing the exponents.
Use a common denominator: $\frac{5}{2} - \frac{4}{3} = \frac{15}{6} - \frac{8}{6} = \frac{7}{6}$ . Thus $a / b = 7/6$ , choice A. This algebra establishes the identity throughout the stated positive-x domain.

Common trap

Do not subtract numerators and denominators separately. Also, adding these exponents would give $23/6$ , not $7/6$ : addition applies to a product of like bases, not this quotient.

Why do the conditions matter?

Positive x keeps both radicals real and the denominator nonzero. Checking only $x = 1$ cannot distinguish exponents because every real power of 1 is 1; the algebra, not a single numerical match, proves the result.

Dolphin question bank #731. Worked explanation reviewed October 8, 2026.

For the same algebra in a different form, try equivalent-expressions practice.

Method reference: OpenStax: radicals and rational exponents.

Dolphin practice questions

Practice your skill

Question 1 of 3Easy

Which expression is equivalent to

x^{3}

x^{4}

$x^{7}$
$x^{1} 2$
$2 x^{7}$
x

Show answer and solution

Correct answer $A . x^{7}$

Solution walkthrough

When multiplying powers with the same base, add the exponents: 3 + 4 = 7.

Question 2 of 3Medium

2^{a}

= 8 and

2^{b}

= 32, what is a + b?

Show answer and solution

Correct answerC. 8

Solution walkthrough

Since 8 = $2^{3}$ and 32 = $2^{5}$ , a = 3 and b = 5, so a + b = 8.

Question 3 of 3Hard

Which expression is equivalent to (

x^{2} y

^-3)^2?

$x^{4} y^{-} 6$
$x^{4} y^{6}$
$x^{2} y^{-} 6$
$x^{4} / y^{-} 6$

Show answer and solution

Correct answer $A . x^{4} y^{-} 6$

Solution walkthrough

Raise each factor to the second power: ( $x^{2}$ )^2 = $x^{4}$ and (y^-3)^2 = y^-6.

Avoid these traps

Common mistakes on this skill

Adding exponents during addition

Exponents add when multiplying powers with the same base, not when adding the powers themselves.

Applying a power to only one factor

A power outside parentheses applies to every factor inside and multiplies existing exponents.

Study plan

How to practice this skill in Dolphin

Identify whether the operation is multiplication, division, or a power of a power.
Apply one exponent rule at a time while keeping the base unchanged where required.
Rewrite negative or fractional exponents and check the result with a simple positive base.

Practice exponent rules in Dolphin SAT

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FAQ

Questions about SAT Exponent Rules Practice

What does a negative exponent mean?

For a nonzero base, a^-n equals 1 divided by a^n.

When do exponents multiply?

They multiply in a power of a power, such as (x^a)^b = x^(ab).