Rewrite expressions involving radicals and rational exponents using the properties of exponents, and simplify square roots and cube roots (A.NR, Numerical Reasoning).

What this topic is asking

The Numerical Reasoning (A.NR) strand asks you to rewrite expressions involving radicals and rational exponents and to put square roots and cube roots in simplest radical form. The big idea is that a radical and a fractional exponent are two notations for the same operation, so once you can convert between them you can use the ordinary exponent rules to simplify. On the Georgia Milestones EOC this is a steady source of one-point selected-response items and short numeric-entry items, and the calculator does not help much because the answers are often exact (a radical or an integer), not a rounded decimal.

Simplifying square roots

A square root is in simplest radical form when the number under the radical has no perfect-square factor other than 1. Use the product rule $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ and pull out the largest perfect square.

If you do not spot the largest factor, peel off perfect squares one at a time: $\sqrt{72} = \sqrt{4 \cdot 18} = 2\sqrt{18} = 2\sqrt{9 \cdot 2} = 2 \cdot 3\sqrt{2} = 6\sqrt{2}$. Both routes reach the same answer; the one-step version is faster on the non-calculator section.

Simplifying cube roots

A cube root is simplest when the radicand has no perfect-cube factor other than 1. The perfect cubes are $1, 8, 27, 64, 125, \dots$.

Cube roots, unlike square roots, are defined for negative radicands: $\sqrt[3]{-8} = -2$, because $(-2)^3 = -8$.

Radicals and rational exponents are the same thing

The single most useful conversion in this topic is:

So $\sqrt{x} = x^{1/2}$, $\sqrt[3]{x} = x^{1/3}$, $\sqrt[3]{x^2} = x^{2/3}$, and $\frac{1}{\sqrt{x}} = x^{-1/2}$. Once in exponent form, the standard rules apply:

How the Milestones examines this topic

• Multiple choice. Choose the simplest radical form of $\sqrt{72}$ or $\sqrt[3]{54}$, with partially simplified distractors.
• Numeric entry. Evaluate a rational-exponent power such as $27^{2/3}$ or $32^{2/5}$, or simplify a product of radicals.
• Drag and drop. Match each radical to its equivalent rational-exponent form.

Why "root first, then power" is faster

When you evaluate $a^{m/n}$, you can take the root before the power or the power before the root; the answer is the same, but the order changes how big the intermediate numbers get. For $8^{2/3}$, taking the root first gives $8^{1/3} = 2$, then squaring gives $4$. Taking the power first would mean computing $8^2 = 64$ and then $\sqrt[3]{64} = 4$, which forces you to recognize that 64 is a perfect cube under pressure. The numbers stay small when you root first, which is exactly what you want on a non-calculator item, and it also makes the perfect-power structure obvious.

Combining radical expressions

You can add or subtract radicals only when they are like radicals (same index and same radicand), the way you combine like terms: $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$, but $3\sqrt{2} + 5\sqrt{3}$ does not combine. Sometimes simplifying first reveals like radicals: $\sqrt{8} + \sqrt{18} = 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}$. You can always multiply radicals with the same index using $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$, so $\sqrt{6}\,\sqrt{10} = \sqrt{60} = 2\sqrt{15}$. Keeping these two rules straight (add only like radicals, multiply freely) prevents the most common simplification errors on the EOC.

Try this

Q1. Simplify $\sqrt{50} + \sqrt{8}$. [1 point]

• Cue. $5\sqrt{2} + 2\sqrt{2} = 7\sqrt{2}$.

Q2. Write $\sqrt[5]{x^3}$ with a rational exponent and evaluate $32^{2/5}$. [2 points]

• Cue. $\sqrt[5]{x^3} = x^{3/5}$; $32^{2/5} = \left(\sqrt[5]{32}\right)^2 = 2^2 = 4$.