Exponent Rules and Radicals | SAT Math Practice

The key idea

Multiplying powers adds exponents, dividing subtracts, and a power of a power multiplies. A radical is a fractional exponent:

n x^{m} = x^{m / n}

. To compare different bases, rewrite them with one base first.

The trap

Adding exponents when raising a power to a power, or writing $1 6^{x - 1}$ as $2^{4 x - 1}$ .

Can Desmos do it?

For an expression in $x$ alone, graphing two forms shows whether they match. With two variables tied by a condition, rewriting the exponents is the only reliable route.

College Board skill: Equivalent expressions · Advanced Math

Try one at each level

One question from each level. The drill has 12 at every level, and moves you up after four right in a row.

Level 1 · Warm-up

For

x > 0

, the expression

\frac{( x ^{5} ) ^{3} \cdot x ^{1}}{x ^{8}}

is equivalent to

x^{k}

. What is the value of

k

Show the answer

Answer: 8A power of a power multiplies exponents:

(x^{5})^{3} = x^{15}

. Multiplying adds and dividing subtracts:

15 + 1 - 8 = 8

Level 2 · Test level

For

x > 0

, the expression

4 x^{7} \cdot x^{1}

is equivalent to

x^{k}

. What is the value of

k

Show the answer

Answer: 11/4Write the radical as a power:

4 x^{7} = x^{7/4}

. Then add exponents:

\frac{7}{4} + 1 = \frac{11}{4}

Level 3 · Beyond the test

2 x - 3 y = - 3

, what is the value of

\frac{4 ^{x + 1} \cdot 8 ^{y}}{1 6 ^{x + 1}}

Show the answer

Answer: 2Rewrite with base 2:

4^{x + 1} = 2^{2 x + 2}

8^{y} = 2^{3 y}

1 6^{x + 1} = 2^{4 x + 4}

. The total exponent is

(2 x + 2) + 3 y - (4 x + 4) = - (2 x - 3 y) - 2 = 1

. The value is

2^{1} = 2

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