SAT Exponents & Radicals: Complete Guide + Practice

Exponent and radical rules appear across SAT Algebra and Advanced Math.

You should know the rules well enough that you do not need to rediscover them under time pressure. See the rest of our SAT Math guides for other Advanced Math topics.

Product rule

[ x^a\cdot x^b=x^{a+b} ]

Example:

[ x^3x^5=x^8 ]

Quotient rule

[ \frac{x^a}{x^b}=x^{a-b} ]

Example:

[ \frac{x^9}{x^4}=x^5 ]

Power of a power

[ (x^a)^b=x^{ab} ]

Example:

[ (x^3)^4=x^{12} ]

Negative exponent

[ x^{-a}=\frac{1}{x^a} ]

Example:

[ 2^{-3}=\frac18 ]

Zero exponent

For nonzero x:

[ x^0=1 ]

Fractional exponents

[ x^{1/2}=\sqrt{x} ]

[ x^{1/3}=\sqrt[3]{x} ]

[ x^{m/n}=\sqrt[n]{x^m} ]

Example:

[ 16^{3/4}=(16^{1/4})^3=2^3=8 ]

Simplifying radicals

[ \sqrt{72}=\sqrt{36\cdot2}=6\sqrt2 ]

Adding radicals

You can combine like radicals:

[ 3\sqrt5+2\sqrt5=5\sqrt5 ]

But:

[ \sqrt2+\sqrt3 ]

does not simplify.

Radical equations

Example:

[ \sqrt{x+5}=4 ]

Square both sides:

[ x+5=16 ]

[ x=11 ]

Always check answers after squaring because squaring can create extraneous solutions. Extraneous solutions show up in quadratic equations too, especially when factoring creates extra roots.

Original practice

1

Simplify:

[ x^4x^7 ]

2

Simplify:

[ \frac{a^{10}}{a^3} ]

3

Evaluate:

[ 81^{1/2} ]

4

Evaluate:

[ 27^{2/3} ]

5

Solve:

[ \sqrt{x-1}=5 ]

Answers

1. (x^{11})
2. (a^7)
3. 9
4. 9
5. 26

Common traps

• adding exponents when taking a power of a power;
• forgetting negative exponent means reciprocal;
• distributing square roots incorrectly over addition;
• failing to check radical-equation solutions.

These same instincts carry over to factoring and polynomials, where spotting structure matters just as much.

Bottom line

Exponent rules should become automatic.

They show up inside bigger SAT questions, where hesitation can cost both time and accuracy.

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