1 · The three exponent rules
With the same base: multiplying adds the exponents, dividing subtracts them, and a power of a power multiplies them. x³ · x⁴ = x⁷, x⁹ ÷ x³ = x⁶, and (x²)⁵ = x¹⁰.
The rules only work with the same base. 2³ · 3⁴ cannot be combined into one power.
2 · Zero and negative exponents
Any nonzero number to the power 0 is 1: 7⁰ = 1. A negative exponent means "one over": 2⁻³ = 1/2³ = 1/8.
A negative exponent never makes the number negative. 5⁻² is 1/25, a small positive number.
3 · Powers of a product
Raise every factor inside the parentheses: (3x²)³ = 3³ · x⁶ = 27x⁶. Writing 9x⁶ (3 × 3) or 3x⁶ (leaving the 3 alone) are the usual slips.
4 · Rewrite with the same base
Many questions become easy once every number is a power of the same base. 8² = (2³)² = 2⁶, and 4¹⁴ = (2²)¹⁴ = 2²⁸.
To solve 3ⁿ × 9 = 3⁷, write 9 as 3²: 3ⁿ⁺² = 3⁷, so n + 2 = 7 and n = 5.
5 · Simplifying square roots
Pull out the largest perfect square: √72 = √36 · √2 = 6√2. Multiplying works the same way: √6 × √15 = √90 = √9 · √10 = 3√10.
Only like roots add: √50 + √8 = 5√2 + 2√2 = 7√2. But √9 + √16 = 3 + 4 = 7, not √25 = 5, so never add the numbers under the roots.
6 · Estimating roots
Find the perfect squares on each side. 81 < 90 < 100, so √90 is between 9 and 10. Since 90 is a little past the middle of 81 and 100, √90 is a little less than 9.5 (it is about 9.49).
7 · Scientific notation
A number between 1 and 10 times a power of 10. 420,000 = 4.2 × 10⁵ and 0.00036 = 3.6 × 10⁻⁴.
Multiply the front numbers and add the exponents, then fix the front number if it is 10 or more: (3 × 10⁴)(5 × 10³) = 15 × 10⁷ = 1.5 × 10⁸.
8 · Rational and irrational numbers
A rational number can be written as a fraction of two integers. That includes whole numbers, decimals that end, and decimals that repeat: 0.272727… = 27/99 = 3/11.
Square roots of numbers that are not perfect squares, like √2 or √20, and π are irrational. √49 = 7 is rational.
9 · Factors, GCF, and LCM
Break numbers into primes: 360 = 2³ · 3² · 5. The number of factors is (3 + 1)(2 + 1)(1 + 1) = 24, because each factor uses 2 zero to three times, 3 zero to two times, and 5 zero or one time.
The GCF takes the lower power of each shared prime; the LCM takes the higher power of every prime. For 48 = 2⁴ · 3 and 72 = 2³ · 3², GCF = 2³ · 3 = 24 and LCM = 2⁴ · 3² = 144. Also, GCF × LCM = 48 × 72.
10 · Remainders
If n leaves a remainder of 4 when divided by 7, then 3n leaves 3 × 4 = 12 extra, and 12 is one more 7 plus 5. Check with a real number: n = 11 gives 3n = 33 = 4 × 7 + 5.
A remainder is always less than the number you divide by. If your remainder is too big, take out more groups.
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