Exponents & roots
Powers, the rules for combining them, and square roots.
In your head, jot if needed · no calculator why?
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the idea
A power is repeated multiplication: 2⁵ means five 2s multiplied together, which is 32. A square root undoes squaring: √49 is 7 because 7 × 7 = 49. A cube root, written ∛, undoes cubing.
more of the idea
Powers of the same base combine by rules that come from counting factors. x³ × x² is five x's multiplied together, so x⁵. Dividing cancels factors, so exponents subtract. Any nonzero number to the power 0 is 1.
techniques
Build a power step by step
- Start at 1 and multiply by the base as many times as the exponent says.
- A negative base: an even exponent gives a positive answer, an odd one a negative.
- The exponent 0 always gives 1.
(−2)⁵
- Without the sign: 2, 4, 8, 16, 32.
- Five negative factors is an odd count, so the answer is negative.
Answer: −32
Count the factors
- Multiplying powers: add the exponents.
- Dividing: subtract the exponent you divide by.
- A power of a power: multiply the exponents.
(x⁴ × x⁶) ÷ x³ = xⁿ. What is n?
- Multiplying adds: 4 + 6 = 10.
- Dividing subtracts: 10 − 3 = 7.
Answer: n = 7
Pull out the biggest square
- Know the squares up to 20 × 20 and the cubes up to 10 × 10 × 10.
- For a√b, find the largest square that divides the number.
- Its root goes outside as a. What is left stays under the root as b.
- Check: b should have no square factor except 1.
Write √200 as a√b in simplest form. What is a?
- Squares that divide 200: 4, 25 and 100.
- The largest is 100, and 200 = 100 × 2.
- √100 is 10, so √200 is 10√2.
Answer: 10
watch out for
- Multiplying the base by the exponent: 2⁷ is 128, not 14.
- Halving instead of taking the root. √196 is 14, because 14 × 14 is 196.
- Multiplying exponents when you multiply powers. x⁵ × x³ is x⁸. Multiply exponents only for a power of a power.
- Stopping early on a root. √72 is 3√8, but 8 still holds the square 4, so keep going to 6√2.
practice
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Powers
worked example
7³
- Start at 1 and multiply by 7, 3 times: 7, 49, 343.
Answer: 343
Square & cube roots
worked example
∛729
- 9 × 9 × 9 = 729, so ∛729 = 9.
Answer: 9
Exponent rules
worked example
(x² × x⁵) ÷ x³ = xⁿ. What is n?
- Multiplying adds exponents: 2 + 5 = 7.
- Dividing subtracts: 7 − 3 = 4.
Answer: 4
Simplify a square root
worked example
Write √162 as a√b in simplest form. Enter a, b.
- Find the largest square factor: 162 = 81 × 2, and √81 = 9.
- So √162 = 9√2, and 2 has no square factor other than 1.
Answer: (9, 2)