Solve exponential equations by matching bases and radical equations by isolating and squaring, checking for extraneous solutions (Algebra).

What this topic is asking

The ACT includes equations with the variable in an exponent (exponential) or under a radical (root). Each has a standard method: match bases for exponentials, and isolate and square for radicals, with a mandatory check because squaring can create false solutions.

Exponential equations by matching bases

When both sides can be written with the same base, the exponents must be equal.

Knowing the small powers of 2, 3, 5 and 10 lets you spot the common base quickly.

Radical equations: isolate, square, check

A radical equation has the variable under a root.

The check is not optional: squaring is not reversible, so it can produce values that satisfy the squared equation but not the original.

Why extraneous solutions appear

Squaring both sides can turn a false statement into a true one. For example, $\sqrt{x} = -2$ has no solution, because a principal square root is never negative; but squaring gives $x = 4$, a value that fails the original ($\sqrt{4} = 2 \neq -2$). So $x = 4$ is extraneous. Whenever you square, you may have widened the solution set, which is exactly why you substitute each candidate back into the original equation and keep only those that work.

Equations with rational exponents

A rational exponent combines the two ideas: $x^{2/3} = 4$ has the variable raised to a fractional power, which is a root and a power together. Solve by raising both sides to the reciprocal power: raise to the $\frac{3}{2}$ power, so $\left(x^{2/3}\right)^{3/2} = 4^{3/2}$, giving $x = 4^{3/2} = (\sqrt{4})^{3} = 2^{3} = 8$. Reading $x^{2/3}$ as "cube root, then square" and undoing it with the reciprocal exponent turns a hard-looking equation into two clean steps. As with radicals, check the result, since even powers can admit a sign you must verify against the original.

Negative and zero exponents in equations

You may also meet an equation like $2^{x} = \frac{1}{8}$, where the right side is less than 1, signalling a negative exponent. Rewrite $\frac{1}{8} = 2^{-3}$, so $2^{x} = 2^{-3}$ and $x = -3$. Recognising that fractions correspond to negative exponents, and that any nonzero base to the zero power is 1, keeps base-matching reliable even when the target is a fraction.

Why these methods are reliable

Both equation types reduce to skills you already have once the exponent or radical is removed: matching bases turns an exponential equation into a simple linear one, and squaring turns a radical equation into a linear or quadratic one. The discipline that separates a correct answer from a wrong one is rewriting to a common base accurately, and always checking radical solutions in the original equation to throw out extraneous values.

Try this

Q1. Solve $5^{x} = 125$. [1 point]

• Cue. $125 = 5^{3}$, so $x = 3$.

Q2. Solve $\sqrt{2x - 1} = 5$. [1 point]

• Cue. Square: $2x - 1 = 25$, so $x = 13$. Check: $\sqrt{25} = 5$, valid.