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Exponent rules

SAT Math Review · Advanced Math

Exponent rules, including rational exponents

Add, subtract, multiply — and read a fraction in the exponent as a root over a power.

For x > 0, which expression is equivalent to the given expression?

√x³∜x

Three rules cover every "which expression is equivalent" item.

Same base, or nothing happens. These three are the whole toolkit.

When you	The rule	On an SAT line
multiply like bases	a^m · aⁿ = a^m+n	x⁵ · x² = x⁷
divide like bases	a^m ÷ aⁿ = a^m−n	x⁹ ÷ x⁴ = x⁵
raise a power to a power	(a^m)ⁿ = a^mn	(x³)⁵ = x¹⁵

A power outside a bracket reaches every factor inside it, the number included.

(3p²q)³

three factors inside

3³ = 27

the 3 is a factor, not a spectator

(p²)³ = p⁶, q³ = q³

2 · 3 = 6

= 27p⁶q³

nothing left inside

Multiplying is not the same as raising

x⁵ · x² = x⁷, but (x⁵)² = x¹⁰. One of those two values is always sitting in the choices. Decide which operation the expression actually shows before you touch the exponents.

A zero power is 1, and a negative power is a location.

The subtraction rule forces both — nobody chose them.

2³ ÷ 2³ = 8 ÷ 8 = 1

anything over itself

2³ ÷ 2³ = 2³⁻³ = 2⁰

the same division, by the rule

so 2⁰ = 1

two names for one number

2³

2²

2¹

2⁰

2⁻¹

2⁻²

each step down halves — the pattern never breaks

x⁻² is not negative

It is not −x², and it is never a negative number when x is positive. A negative exponent says which side of the fraction bar the factor belongs on: x⁻² = 1x². Move it across and the sign turns positive.

A fraction in the exponent: the bottom is the root, the top is the power.

x^m/n = ⁿ√(x^m) — denominator down at the root sign, numerator up on the x.

x^3/2 root 2 power 3 = √x³

Exponent form	Radical form	At x = 64
x^1/2	√x	8
x^1/3	³√x	4
x^2/3	³√x² = (³√x)²	16
x^3/2	√x³ = (√x)³	512
x^−1/2	1√x	0.125

Root first or power first — the value is the same, and root first keeps the numbers small: 64^2/3 = (³√64)² = 4² = 16, which beats squaring 64 and then taking a cube root of 4,096.

Say it out loud once

"Three over two means square root of x cubed." The denominator lives down in the root — same place it sits in the fraction. Flipping the two numbers is the most expensive mistake on this topic, and the flipped value is always one of the choices.

Convert to exponent form, then use the ordinary rules.

Radicals do not have their own arithmetic. Turn them into exponents and the three rules from step 1 take over.

√x³∜x

√x³ = x^3/2

root 2 below, power 3 above

∜x = x^1/4

root 4 below, power 1 above

x^3/2 ÷ x^1/4 = x^{3/2 − 1/4}

dividing subtracts

32 − 14 = 64 − 14 = 54

common denominator first

= x^5/4 = ∜(x⁵)

back to radical form

√x³ ÷ ∜x = x^5/4 = ∜(x⁵)

The distractor is the addition

32 + 14 = 74 gives ∜(x⁷), and it is on the list. A fraction bar between the two pieces means subtract, however unfamiliar the fractions look.

Deciding which of four forms matches.

Put every choice into exponent form on one base, then compare a single number.

(³√a)⁵ · √a , a > 0

(³√a)⁵ = a^5/3

root 3, power 5

√a = a^1/2

root 2, power 1

53 + 12 = 106 + 36 = 136

multiplying adds

= a^13/6

the target

Choice	As a^k	Match?
⁶√a¹³	a^13/6	yes — same expression, different clothes
a²·⁶√a	a^{2 + 1/6} = a^13/6	yes — and this is how the SAT usually prints it
a^5/6	a^5/6	no — that is 53 × 12, multiplying the exponents instead of adding
⁶√a⁵	a^5/6	no — the same slip, dressed as a radical

Confirm it with the calculator

You have Desmos the whole section. Put a = 64: the expression is (³√64)⁵ · √64 = 4⁵ · 8 = 1024 · 8 = 8192, and 64^13/6 = 8192 as well. One substitution kills three wrong choices in about fifteen seconds — do it whenever the algebra felt shaky.

Different bases? Rewrite them as powers of one base.

The rules need a common base. 4, 8, 16 and 32 are all powers of 2; 9, 27 and 81 are all powers of 3.

4 = 2² 8 = 2³ 16 = 2⁴ 9 = 3² 27 = 3³

27^x · 9^2x = 3^kx

27^x = (3³)^x = 3^3x

power to a power: multiply

9^2x = (3²)^2x = 3^4x

2 · 2x = 4x

3^3x · 3^4x = 3^7x

now the bases match, so add

k = 7

compare the exponents

Once both sides are a power of the same base, the exponents themselves must be equal.

16^x−1 = 4^x+3

both sides are powers of 2

2^4(x−1) = 2^2(x+3)

16 = 2⁴, 4 = 2²

4x − 4 = 2x + 6

set the exponents equal

x = 5

2x = 10

Check it: 16⁴ = 65,536 and 4⁸ = 65,536 ✓. A rational exponent works the same way — 8^x = 4 becomes 2^3x = 2², so x = 23.

Check: substitute one friendly number.

Pick a value that makes the roots come out whole — 64 works for halves, thirds and sixths.

the opener at x = 16 √16³ = √4096 = 64 the numerator

the denominator ∜16 = 2 16 = 2⁴

divide 64 ÷ 2 = 32 the original expression

our answer ∜(16⁵) = ∜1,048,576 = 32 a match ✓

the tempting ∜(x⁷) ∜(16⁷) = 2⁷ = 128 not 32 — ruled out ✓

Remember the flow

Exponent form · Same base · Add, subtract or multiply · Back to the form the choices use

Denominator = root, numerator = power. Never the other way round.

Now you try

Practice problem

For x > 0, which expression is equivalent to the given expression?

(16x⁸)^3/4 · x⁻²

Hint: clear the bracket first and do the number separately from the letter — 16^3/4 means fourth root, then cube. Only then combine with x⁻², remembering that a negative exponent subtracts.

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