Math

Exponents and Radicals

5 min readMedium5-question drill

Exponent and radical questions reward students who know the rules — there are only about eight you need, and once they're memorized these problems become 30-second wins.

Where this lesson fits

Before

Polynomials→

Exponents and Radicals

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→Exponential Functions →Quadratic Equations

The eight exponent rules to memorize:

Product: $a^m \cdot a^n = a^{m+n}$ . Example: $x^3 \cdot x^4 = x^7$ .
Quotient: $\frac{a^m}{a^n} = a^{m-n}$ . Example: $\frac{x^7}{x^3} = x^4$ .
Power of a power: $(a^m)^n = a^{mn}$ . Example: $(x^2)^3 = x^6$ .
Power of a product: $(ab)^n = a^n b^n$ .
Negative exponent: $a^{-n} = \frac{1}{a^n}$ . Example: $x^{-2} = \frac{1}{x^2}$ .
Zero exponent: $a^0 = 1$ (any nonzero base).
Fractional exponent: $a^{m/n} = \sqrt[n]{a^m}$ . Example: $x^{1/2} = \sqrt{x}$ .
Power of a quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

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The eight exponent rules — your complete cheat sheet

Rule	Formula	Example
Product	$a^m \cdot a^n = a^{m+n}$	$x^3 \cdot x^4 = x^7$
Quotient	$a^m / a^n = a^{m-n}$	$x^7 / x^3 = x^4$
Power of power	$(a^m)^n = a^{mn}$	$(x^2)^3 = x^6$
Power of product	$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
Negative exponent	$a^{-n} = 1/a^n$	$x^{-2} = 1/x^2$
Zero exponent	$a^0 = 1$	$5^0 = 1$
Fractional exponent	$a^{m/n} = \sqrt[n]{a^m}$	$x^{1/2} = \sqrt{x}$
Power of quotient	$(a/b)^n = a^n/b^n$	$(x/2)^3 = x^3/8$

Radicals (roots). A square root undoes squaring; a cube root undoes cubing.

$\sqrt{a^2} = |a|$ (absolute value, since both $a$ and $-a$ square to $a^2$ ).
$\sqrt[3]{a^3} = a$ (cube roots preserve sign).

Radicals as fractional exponents:

$\sqrt{x} = x^{1/2}$
$\sqrt[3]{x} = x^{1/3}$
$\sqrt{x^3} = x^{3/2}$

Convert radicals to fractional exponents to apply exponent rules — much easier.

Simplifying radicals: factor out perfect squares.

$\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$ .

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Simplifying radicals — extract perfect squares

Input	Factor with perfect square	Simplified
$\sqrt{72}$	$\sqrt{36 \cdot 2}$	$6\sqrt{2}$
$\sqrt{50}$	$\sqrt{25 \cdot 2}$	$5\sqrt{2}$
$\sqrt{18}$	$\sqrt{9 \cdot 2}$	$3\sqrt{2}$
$\sqrt{x^5}$	$\sqrt{x^4 \cdot x}$	$x^2 \sqrt{x}$

Solving exponential equations: if both sides have the same base, set the exponents equal.

$2^{x+1} = 2^{2x-3} \Rightarrow x + 1 = 2x - 3 \Rightarrow x = 4$ .

If the bases differ, rewrite to match: $4^x = 8 \Rightarrow (2^2)^x = 2^3 \Rightarrow 2^{2x} = 2^3 \Rightarrow 2x = 3 \Rightarrow x = 1.5$ .

Common SAT trap: $a^m + a^n$ is NOT $a^{m+n}$ . The product rule applies to multiplication, not addition. $2^3 + 2^4 = 8 + 16 = 24$ , not $2^7 = 128$ .

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Quick check

Identify the rule that applies (product / quotient / power-of-power / fractional / negative). Then plug in carefully — exponents are unforgiving on sign and arithmetic errors.

Which expression is equivalent to 5^5 · 5^1?

5^5

10^6

5^4

5^6

Worked examples

Example 1

Simplify: $\frac{x^5 \cdot x^{-2}}{x^4}$ .

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Example 2

Solve for $x$ : $9^{x+1} = 27^{x-2}$ .

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Common pitfalls

Adding $a^m + a^n$ as $a^{m+n}$

Product rule applies only to multiplication: $a^m \cdot a^n = a^{m+n}$ . For addition, you can't combine. $2^3 + 2^4 = 8 + 16 = 24$ , NOT $2^7$ . Misapplying this rule is the most common exponent error.

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Forgetting the absolute value with even roots

$\sqrt{x^2} = |x|$ , not $x$ . If $x = -3$ , $\sqrt{(-3)^2} = \sqrt{9} = 3$ , not $-3$ . Even-degree roots always return non-negative values.

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Distributing exponents over addition

$(a + b)^2 \neq a^2 + b^2$ . You have to FOIL out: $(a + b)^2 = a^2 + 2ab + b^2$ . The exponent only distributes over multiplication: $(ab)^2 = a^2 b^2$ .

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Confusing $x^{1/2}$ with $\frac{1}{x^2}$

Fractional exponent = root: $x^{1/2} = \sqrt{x}$ . Negative exponent = reciprocal: $x^{-2} = \frac{1}{x^2}$ . Don't mix them up.

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Key takeaways

Same-base multiplication: add exponents. Same-base division: subtract.
Power of a power: multiply exponents.
Negative exponent flips to the reciprocal: $a^{-n} = \frac{1}{a^n}$ .
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Fractional exponent $a^{m/n}$ = $n$ -th root of $a^m$ . Convert radicals to fractional exponents to apply rules.
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To solve $a^? = a^?$ : rewrite both sides with the same base, then set exponents equal.
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Tracks your progress across lessons.

Try it yourself

5 practice questions on Exponents and Radicals, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Before

Polynomials→

Exponents and Radicals

You are here

→Exponential Functions →Quadratic Equations