Work out the value of 125−2/3.
- The minus sign means reciprocal: 125−2/3 = 1/(1252/3).
- The 3 on the bottom means cube root: the cube root of 125 is 5.
- The 2 on the top means square: 52 = 25.
Answer: 1/25
Powers, roots, indices and standard form
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 (index) tells you how many times a number is multiplied by itself. A root undoes a power: √49 = 7 because 72 = 49.
Indices do not have to be positive whole numbers. A negative index means "one over" (a reciprocal). A fractional index means a root: the bottom of the fraction is the root and the top is the power. The laws of indices only work when the base is the same, so first write numbers such as 9 or 32 as powers of 3 or 2.
Standard form writes very large or very small numbers as A × 10n, where A is at least 1 and less than 10.
Rules to remember
- Multiply: add the indices (am × an = am + n). Divide: subtract them (am ÷ an = am − n).
- Power of a power: multiply the indices ((am)n = amn). Everything in the bracket is raised: (2x)3 = 8x3.
- a0 = 1 and a−n = 1/an.
- a1/n is the nth root of a. am/n = (nth root of a)m: take the root first, then the power.
- Standard form: A × 10n with 1 ≤ A < 10. n is positive for large numbers and negative for numbers less than 1.
- To multiply in standard form, multiply the A values and add the powers of 10. To divide, divide the A values and subtract the powers. Then adjust so that A is between 1 and 10.
- To estimate a root, find the square numbers on each side: √40 is between √36 = 6 and √49 = 7.
Common mistake
Thinking a negative index makes the answer negative. 2−3 is 1/8, not −8: the minus sign means reciprocal.
Worked example
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