Squares, Cubes & Roots of Fractions — Grade 8 Maths (CAPS)

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

a/b² = a²/b² and a/b³ = a³/b³ (b ≠ 0)

Worked ExampleCalculate: (2/3)² and (3/4)³

1
(2/3)² = 2²/3² = 4/9.
Square the top and square the bottom.
2
(3/4)³ = 3³/4³ = 27/64.
Cube the top and cube the bottom.

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Square both parts, not just one. (⅔)² is 4/9, not 4/3. The whole fraction is being squared, so the denominator is squared too.

2Square roots and cube roots of fractions

√(a/b) = (√a)/(√b) and ∛(a/b) = (∛a)/(∛b)
For the square root, a ≥ 0 and b > 0; for the cube root, b ≠ 0.

Worked ExampleCalculate: √(9/16) and ∛(8/27)

1
√(9/16) = (√9)/(√16) = 3/4.
Root the top and root the bottom.
2
∛(8/27) = (∛8)/(∛27) = 2/3.
Because 2 × 2 × 2 = 8 and 3 × 3 × 3 = 27.

3Mixed numbers must be converted first

Worked ExampleCalculate: (11/2)²

1
Convert first: 11/2 = 3/2.
You cannot square the whole and the fraction separately.
2
(3/2)² = 9/4.
Square top and bottom.
3
= 21/4.
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.