reframing radicals and exponents

An SAT Math micro-topic under Radicals and rational exponents (Advanced Math). Free to read — no account needed.

To simplify a radical, split what is inside into a part that forms complete sets and a leftover part.
Each complete set comes out of the radical as one factor, and whatever is left stays under it.

For a square root, a complete set is a pair, since

\sqrt{a^2} = a

.
Take

\sqrt{72}

: since

72 = 36 \times 2

and

36

is a perfect square,

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

For a cube root, a complete set is a triple, since

\sqrt[3]{a^3} = a

.
Take

\sqrt[3]{24}

: since

24 = 8 \times 3

and

8 = 2^3

\sqrt[3]{24} = \sqrt[3]{8} \times \sqrt[3]{3} = 2\sqrt[3]{3}

Variables work the same way.
For a square root,

\sqrt{x^5} = \sqrt{x^4} \times \sqrt{x} = x^2\sqrt{x}

, since

x^4

is two complete pairs.
Pull out as many complete sets as you can, and leave the rest inside.

Worked examples

Simplify

\sqrt{50}

.
Split off a perfect square:

50 = 25 \times 2

, and

25 = 5^2

.
So

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

Simplify

\sqrt{50x^5}

.
Break it into complete pairs and leftovers:

50x^5 = (25 \times x^4) \times (2 \times x)

.
The pairs come out as

5

and

x^2

, so

\sqrt{50x^5} = 5x^2\sqrt{2x}

Simplify

\sqrt[3]{24x^4}

.
Look for complete triples:

24 = 2^3 \times 3

and

x^4 = x^3 \times x

.
The

2^3

and

x^3

come out as

2

and

x

, leaving

\sqrt[3]{24x^4} = 2x\sqrt[3]{3x}

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reframing radicals and exponents

Worked examples

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