Squares, Cubes & Roots of Fractions

Squaring and cubing fractions, and finding square roots and cube roots of fractions — the rule is simply 'do it to the top and to the bottom', with worked examples, a worksheet with memo and a quiz.

This is one of the friendliest topics in Grade 8, because there is a single rule that covers all four operations: whatever you do to the fraction, you do to the top and the bottom separately.

1Squaring and cubing a fraction

ab² = a²b² and ab³ = a³b³

Worked ExampleCalculate: (23)² and (34)³

1
(23)² = 2²3² = 49.
Square the top and square the bottom.
2
(34)³ = 3³4³ = 2764.
Cube the top and cube the bottom.

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Square both parts, not just one. (⅔)² is 49, not 43. The whole fraction is being squared, so the denominator is squared too.

2Square roots and cube roots of fractions

ab = ab and ab = ab

Worked ExampleCalculate: 916 and 827

1
916 = 9 / 16 = 34.
Root the top and root the bottom.
2
827 = 8 / 27 = 23.
Because 2×2×2 = 8 and 3×3×3 = 27.

3Mixed numbers must be converted first

Worked ExampleCalculate: (112)²

1
Convert first: 112 = 32.
You cannot square the whole and the fraction separately.
2
(32)² = 94.
Square top and bottom.
3
= 214.
Convert back to a mixed number if the question used one.

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It helps to know the first few square numbers (1, 4, 9, 16, 25, 36, 49, 64, 81, 100) and cube numbers (1, 8, 27, 64, 125) by heart — then these questions take seconds.