Powers and Roots

Why Exponents Rule the ACT

Exponents are shorthand for repeated multiplication: $a^n$ means $a$ multiplied by itself $n$ times. On the ACT Math section they show up in pure-algebra simplifications, in scientific notation, in growth models, and inside geometry (areas and volumes). A handful of rules handles almost every one of these problems, so learning the rules cold is one of the best return-on-time investments on the whole test.

The Core Exponent Rules

Every exponent question is built from these identities. Memorize them until they are automatic.

Rule	Statement	Example
Product	$a^m \cdot a^n = a^{m+n}$	$x^3 \cdot x^5 = x^8$
Quotient	$\dfrac{a^m}{a^n} = a^{m-n}$	$\dfrac{x^7}{x^2} = x^5$
Power of a power	$(a^m)^n = a^{mn}$	$(x^4)^3 = x^{12}$
Power of a product	$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
Zero exponent	$a^0 = 1$ (for $a \ne 0$)	$17^0 = 1$

Notice the pattern: multiplying like bases adds exponents, dividing subtracts them, and a power of a power multiplies them. Mixing these up is the single most common ACT exponent error.

Negative and Rational Exponents

Two extensions unlock the harder questions.

A negative exponent means take the reciprocal:

$a^{-n} = \frac{1}{a^n}$

So $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$. A negative exponent never makes a number negative — it makes it a fraction.

A rational (fractional) exponent is a root:

$a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$

The denominator is the root and the numerator is the power. For example, $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$. Taking the cube root first keeps the numbers small.

Working with Radicals

A radical (or root) undoes a power. The most tested move is simplifying a square root by pulling out perfect-square factors:

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

You can multiply and divide radicals directly, $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$, but you cannot add unlike radicals: $\sqrt{2} + \sqrt{3}$ does not simplify. You can only combine like radicals, the way you combine like terms: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$.

When a radical sits in a denominator, rationalize it by multiplying top and bottom by that radical:

$\frac{6}{\sqrt{3}} = \frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$

Scientific Notation

The ACT uses exponents to write very large or very small numbers as $a \times 10^n$ with $1 \le a < 10$. To multiply, handle the decimals and the powers of 10 separately:

$(3 \times 10^5)(2 \times 10^{-2}) = (3 \cdot 2) \times 10^{5 + (-2)} = 6 \times 10^3$

A Reliable Game Plan

• Same base, multiplied or divided? Add or subtract the exponents.
• A power raised to a power? Multiply the exponents.
• See a negative exponent? Rewrite as a reciprocal before simplifying.
• See a fractional exponent? Denominator is the root, numerator is the power — take the root first.
• Simplifying a root? Factor out the largest perfect square (or cube).

Common Mistakes to Avoid

• Adding exponents when raising a power to a power: $(x^3)^2$ is $x^6$, not $x^5$.
• Treating a negative exponent as a negative number: $2^{-2} = \frac14$, not $-4$.
• Adding unlike radicals: $\sqrt{9 + 16}$ is $\sqrt{25} = 5$, not $3 + 4 = 7$.
• Distributing an exponent over a sum: $(a + b)^2 \ne a^2 + b^2$.