doing math

courses › Pre-Algebra

Exponents & roots

level 18 course

Powers, the rules for combining them, and square roots.

In your head, jot if needed · no calculator why?

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Builds on: Factors & primes (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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.

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

Working out a power like 2⁷ or (−3)³.

  1. Start at 1 and multiply by the base as many times as the exponent says.
  2. A negative base: an even exponent gives a positive answer, an odd one a negative.
  3. The exponent 0 always gives 1.
worked example

Example: (−2)⁵

  1. Without the sign: 2, 4, 8, 16, 32.
  2. Five negative factors is an odd count, so the answer is negative.

Answer: −32

Count the factors

Combining powers of the same base, like x⁵ × x³ or (x²)⁴.

  1. Multiplying powers: add the exponents.
  2. Dividing: subtract the exponent you divide by.
  3. A power of a power: multiply the exponents.
worked example

Example: (x⁴ × x⁶) ÷ x³ = xⁿ. What is n?

  1. Multiplying adds: 4 + 6 = 10.
  2. Dividing subtracts: 10 − 3 = 7.

Answer: n = 7

Pull out the biggest square

Square and cube roots, and writing a root like √72 as a√b.

  1. Know the squares up to 20 × 20 and the cubes up to 10 × 10 × 10.
  2. For a√b, find the largest square that divides the number.
  3. Its root goes outside as a. What is left stays under the root as b.
  4. Check: b should have no square factor except 1.
worked example

Example: Write √200 as a√b in simplest form. What is a?

  1. Squares that divide 200: 4, 25 and 100.
  2. The largest is 100, and 200 = 100 × 2.
  3. √100 is 10, so √200 is 10√2.

Answer: 10

Tips by skill

  • TipPowers: Start at 1 and multiply by the base, exponent times. A negative base flips sign with each factor. The exponent 0 gives 1.
  • TipSquare & cube roots: Ask which number times itself, or three times for ∛, gives the number under the root.
  • TipExponent rules: Same base: multiplying adds exponents, dividing subtracts them, and a power of a power multiplies them.
  • TipSimplify a square root: Find the largest square that divides the number. Its root is a, and what is left under the root is b.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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