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Exponents and roots
Exponents are just a shorthand for multiplying the same number again and again. Three rules handle almost every question, and roots are the same idea run backwards.
An exponent counts the factors
- 2⁵
- 2 × 2 × 2 × 2 × 2
- 32
In 2⁵ the base is 2 and the exponent is 5: five 2s multiplied together. “Squared” means exponent 2 and “cubed” means exponent 3. A root goes the other way: √49 asks which number times itself gives 49.
Never multiply the base by the exponent
3⁴ is 3 × 3 × 3 × 3 = 81, not 3 × 4 = 12. The wrong answer is always among the choices.
How to solve them
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1
Powers worth knowing by heart
Knowing a few powers saves time on every question. Two special cases: any number to the power 1 is itself, and any nonzero number to the power 0 is 1.
Next step: The rulesPowers of 33² · 3³ · 3⁴
9 · 27 · 81
Powers of 1010³
1 followed by 3 zeros: 1,000
Power 0 and 17⁰ and 7¹
1 and 7
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2
Multiply: add the exponents. Divide: subtract. Power of a power: multiply
These rules only work when the bases are the same. They follow from counting factors: x² · x³ is (x · x) · (x · x · x), five x’s in all.
x² · x³x⁵add: 2 + 3x⁷ ÷ x³x⁴subtract: 7 − 3(x²)³x⁶multiply: 2 × 3(2x)³8x³cube both partsNext step: Zero & negative- 2³ · 2²
- 2³⁺²
- 2⁵
- 32
Check: 8 × 4 = 32. -
3
Zero and negative exponents
A negative exponent means “one over”: 2⁻³ = 1/2³. It does not make the number negative. And watch the parentheses with a negative base.
Next step: Square rootsZero exponent9⁰
1
Negative exponent2⁻³
1 ÷ 2³ = 1/8
Negative base(−2)⁴ and −2⁴
16 and −16
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4
Square roots undo squares
Learn the perfect squares up to 15 × 15. For a number that is not a perfect square, find the two perfect squares around it.
√648 × 88√12111 × 1111√16913 × 1313√22515 × 1515Next step: Simplify a rootExampleBetween which two whole numbers is √50?
- 49 < 50 < 64.
- So √49 < √50 < √64, that is, 7 < √50 < 8.
Answer Between 7 and 8 (just above 7)
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5
Simplify a root: pull out a perfect square
Split the number into a perfect square times what is left, using the biggest perfect square you can. The square comes out of the root as its square root.
Next step: Cube roots- √72
- √(36 × 2)
- √36 × √2
- 6√2
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6
Cube roots undo cubes
∛ asks which number times itself three times gives the number inside. Unlike square roots, a cube root of a negative number is fine: it is negative.
Small cubes∛27 and ∛125
3 and 5
Powers of 10∛1,000
10
Negative inside∛−8
−2, because (−2)³ = −8
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Base times exponent
3⁴:
Wrong3 × 4 = 12
Right3 × 3 × 3 × 3 = 81
Adding exponents when adding
x² + x³:
Wrongx⁵
Rightcannot be combined
Multiplying exponents when multiplying
x² · x⁴:
Wrongx⁸
Rightx⁶ (add: 2 + 4)
The root of a sum
√(9 + 16):
Wrong3 + 4 = 7
Right√25 = 5
5 questions at test level
Pick an answer, then read why. No timer, no account.
What is the value of 4³?
Show the answer
C. 4³ = 4 × 4 × 4 = 16 × 4 = 64. 12 multiplies the base by the exponent, and 16 is 4². The value is 64.
Simplify: x⁵ · x³
Show the answer
B. Same base, multiplied: add the exponents, 5 + 3 = 8. x¹⁵ multiplies them, and x² subtracts them. The answer is x⁸.
What is √144 + √25?
Show the answer
B. Take each root first: √144 = 12 and √25 = 5, so 12 + 5 = 17. 13 adds 144 + 25 = 169 first and then takes the root, which is not allowed. The answer is 17.
Simplify: √98
Show the answer
A. 98 = 49 × 2, and 49 is a perfect square: √98 = √49 × √2 = 7√2. 49√2 forgets to take the root of 49. The answer is 7√2.
What is 5⁰ + 2⁻²?
Show the answer
C. 5⁰ = 1 and 2⁻² = 1/2² = 1/4 = 0.25, so 1 + 0.25 = 1.25. 0.25 treats 5⁰ as 0, and −4 makes the negative exponent a negative number. The answer is 1.25.
11 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 11 cards known.
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