Work out ∛125 − √0.36.
- ∛125 = 5 because 53 = 125.
- √0.36 = √(36/100) = 6/10 = 0.6.
- 5 − 0.6 = 4.4.
Answer: 4.4
Powers and roots
Maths · Number. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
A power tells you how many times a number is multiplied by itself: 53 means 5 × 5 × 5, not 5 × 3. Squaring is the power 2 and cubing is the power 3.
A root undoes a power. The square root of 49 is 7 because 72 = 49, and the cube root of 64 is 4 because 43 = 64. The sign √ means the positive square root only.
Negative numbers need care. A negative number squared is positive, but a negative number cubed stays negative, so a negative number has a cube root but no real square root. Learn the squares up to 152 and the cubes up to 53 and 103.
Rules to remember
- a2 = a × a and a3 = a × a × a
- Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
- Cubes: 1, 8, 27, 64, 125 and 103 = 1000
- (−a)2 is positive; (−a)3 is negative, so ∛(−8) = −2
- √(a/b) = √a ÷ √b and ∛(a/b) = ∛a ÷ ∛b
- For decimals use fractions: √0.25 = √(25/100) = 5/10 = 0.5
- To place a root between whole numbers, find the squares on each side: 36 < 40 < 49, so √40 is between 6 and 7.
Common mistake
Multiplying the base by the power, such as writing 25 = 10. It is 2 × 2 × 2 × 2 × 2 = 32.
Worked example
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