Radicals and rational exponents

How the SAT tests this, with worked examples — plus the 8 skills it covers. Free to read, no account needed.

Almost every question here becomes straightforward once you convert radicals into exponents. A root is a fractional power

{\sqrt[n]{x^{m}} = x^{m/n}}

and once everything is written as a power, the ordinary exponent rules apply.

The rules you need are few:

x^a \cdot x^b = x^{a+b}

\frac{x^a}{x^b} = x^{a-b}

, and

(x^a)^b = x^{ab}

. A negative exponent means a reciprocal,

x^{-a} = \frac{1}{x^a}

, and a zero exponent gives

1

Simplifying a radical means pulling out perfect squares, so

\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}

. You can only add or subtract radicals once they share the same radicand, exactly as you can only combine like terms.

The one thing that is not true, and is tested constantly, is splitting a root over addition.

\sqrt{a+b}

is not

\sqrt{a} + \sqrt{b}

. Roots distribute over multiplication and division only.

Worked examples

Simplify

\sqrt{72}

.
Factor out the largest perfect square

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

Write

\sqrt[3]{x^{2}}

as a power of

x

.
The index

3

becomes the denominator and the power

2

becomes the numerator

{\sqrt[3]{x^{2}} = x^{2/3}}

Simplify {
sqrt

18

+
sqrt

8

sqrt

26

}
(B) {5
sqrt

2

}
Roots do not add across a sum, so (A) is wrong. Simplify each first: {
sqrt

18

= 3
sqrt

2

} and {
sqrt

8

= 2
sqrt

2

}. They are now like terms, giving {5
sqrt

2

}, so (B) is correct.

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