Exponents and Radicals

Exponents and Radicals

Exponent and radical questions are some of the most mechanical on the ACT. Nearly every one is solved by applying a short list of rules in the right order, then checking the answer. This chapter gives you those rules, the standard simplifying moves, and the one trap that catches most students: extraneous roots.

The core exponent rules

• Multiply powers with the same base: add the exponents. $x^3 \cdot x^5 = x^8$.
• Divide powers with the same base: subtract the exponents. $x^5 \div x^2 = x^3$.
• Raise a power to a power: multiply the exponents. $(x^3)^2 = x^6$.

When a coefficient is inside the parentheses, it gets the exponent too: $(2x^3)^2 = 2^2 \cdot x^6 = 4x^6$. When rules combine, apply the power rule first, then multiply or divide: $(x^4)^3 \div x^5 = x^{12} \div x^5 = x^7$. A negative exponent means a reciprocal, and it simply adds as a signed number: $x^{-2} \cdot x^5 = x^3$.

Rational exponents connect powers and roots

When defined in the real numbers, $x^{m/n}$ means the $n$th root of $x$ raised to the $m$ power. The denominator of the exponent becomes the root; the numerator stays as the power. So $\sqrt[4]{x^3} = x^{3/4}$, and $27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$. Take the root first; the numbers stay small. With an even root, the radicand must be nonnegative for the result to be a real number.

The same rules apply to fractional exponents: $x^{-2/5} \cdot x^{7/5}$ adds the exponents to get $x^{5/5} = x$. Do not multiply the exponents unless one power is being raised to another.

Worked Example. Solve $x^{3/2} = 27$.

Raise both sides to the reciprocal power $\frac{2}{3}$: $x = 27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$.

Combining and simplifying radicals

A radical simplifies by factoring out the largest perfect square. $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$, since $25$ is the largest perfect-square factor of $50$. Two radicals with the same radicand after simplifying can be combined the way like terms are combined.

Worked Example. Simplify $\sqrt{72}$.

Factor out the largest perfect square: $72 = 36 \cdot 2$, so $\sqrt{72} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$.

For sums, simplify every radical first, then combine: $\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$. Never add the numbers under the radical signs: $\sqrt{12} + \sqrt{27}$ is not $\sqrt{39}$.

Rationalizing denominators

A fraction with a radical in the denominator is not considered fully simplified. Rationalizing multiplies the fraction by a form of $1$ built from that radical, which clears it from the denominator without changing the fraction's value.

Worked Example. Rationalize the denominator of $\frac{3}{\sqrt{2}}$.

Multiply by $\frac{\sqrt{2}}{\sqrt{2}}$: $\frac{3\sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{3\sqrt{2}}{2}$.

Do not forget the final division: $\frac{4}{\sqrt{5}}$ becomes $\frac{4\sqrt{5}}{5}$, not $4\sqrt{5}$. If the denominator is a sum such as $4 + \sqrt{2}$, multiply by its conjugate $4 - \sqrt{2}$; the denominator becomes $16 - 2 = 14$.

Solving radical and exponential equations

To solve a radical equation, isolate the radical, then square both sides (or raise to the matching power). Always check every candidate in the original equation, because squaring can introduce an extraneous root.

Worked Example. Solve $\sqrt{x - 1} = x - 3$.

Squaring gives $x - 1 = x^2 - 6x + 9$, so $x^2 - 7x + 10 = 0$ and $x = 5$ or $x = 2$. Check $x = 5$: $\sqrt{4} = 2$ and $5 - 3 = 2$, which works. Check $x = 2$: $\sqrt{1} = 1$ but $2 - 3 = -1$, so $x = 2$ is extraneous. The only solution is $x = 5$.

For an unknown exponent, rewrite both sides with the same base and set the exponents equal. $4^x = \frac{1}{16}$ becomes $4^x = 4^{-2}$, so $x = -2$. And $3^{x-2} = 81 = 3^4$ gives $x - 2 = 4$, so $x = 6$.

Where this appears as a model

Repeated growth described in words is an exponential model. A bacteria population starting at $100$ and doubling every hour follows $P(t) = 100 \cdot 2^t$, giving $P(3) = 800$ after $3$ hours.

Strategy summary

• Same base: add exponents to multiply, subtract to divide, multiply for power of a power.
• Radicals: pull out the largest perfect square; combine only like radicals.
• Rational exponent: denominator is the root, numerator is the power.
• Radical equations: isolate, square, solve, then check every answer.