Simplify square-root and cube-root radical expressions involving numerical and monomial radicands, and convert between radical notation and rational-exponent notation (A.EO.3).

What this topic is asking

A.EO.3 asks you to simplify radical expressions (square roots and cube roots, with numerical and monomial radicands) and to convert between radical notation and rational-exponent notation. On the Virginia Algebra I SOL these are Expressions and Operations items: simplify a radical, write a radical in exponent form (or the reverse), and recognize equivalent forms. They show up as multiple choice, fill-in-the-blank, and matching.

Simplifying square roots

A square root is fully simplified when its radicand has no perfect-square factors other than $1$. The method is to factor out the largest perfect square:

The perfect squares to recognize are $4, 9, 16, 25, 36, 49, 64, 81, 100$. If you remove a smaller square than the largest, the radical is not done yet, so check whether the remaining radicand still has a square factor.

The two properties that justify this are:

• Product property. $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ for $a, b \ge 0$.
• Quotient property. $\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}$ for $a \ge 0, b > 0$.

Radicals with variables

For a variable radicand, pair the factors. Since $\sqrt{x^2} = x$ (for $x \ge 0$ in Algebra I), an even power comes out cleanly and an odd power leaves one factor inside:

Rational exponents

A rational (fractional) exponent is another way to write a radical. The rule is

The denominator $n$ is the index (which root), and the numerator $m$ is the power. So:

• $x^{1/2} = \sqrt{x}$ (square root).
• $x^{1/3} = \sqrt[3]{x}$ (cube root).
• $x^{2/3} = \sqrt[3]{x^2}$ (cube root of $x$ squared).
• $x^{3/2} = \sqrt{x^3}$ (square root of $x$ cubed).

Rational exponents obey the same laws of exponents as integer exponents, which is the whole point of the notation: $x^{1/2} \cdot x^{1/2} = x^1 = x$, exactly as $\sqrt{x} \cdot \sqrt{x} = x$ should be.

Converting and recognizing equivalents

SOL items often hand you one form and ask for an equivalent. Translate using the rule and simplify if possible: $\sqrt[3]{8x^6} = (8x^6)^{1/3} = 8^{1/3} (x^6)^{1/3} = 2x^2$. Going the other way, $16^{3/4} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$. Taking the root first (when the base is a perfect power) keeps the numbers small and avoids large intermediate values.

How the SOL examines this topic

• Multiple choice. Identify the simplified radical, or the equivalent rational-exponent or radical form.
• Fill-in-the-blank. Type a fully simplified radical such as $6\sqrt{2}$, or evaluate an expression with a rational exponent.
• Matching. Pair radical expressions with their simplified or rational-exponent equivalents.

A clarifying idea: a radical and a rational exponent are the same operation written two ways, so you can always switch to whichever form makes the problem easier, then convert back.

Try this

Q1. Simplify $\sqrt{98}$. [1 point]

• Cue. $\sqrt{49 \cdot 2} = 7\sqrt{2}$.

Q2. Write $\sqrt[3]{x^2}$ with a rational exponent. [1 point]

• Cue. $x^{2/3}$ (root $3$ is the denominator, power $2$ is the numerator).