AFOQT Math Practice

Exponents and roots

Rule	Example
Multiply the same base: add the exponents. $a^{m} \cdot a^{n} = a^{m + n}$	$x^{3} \cdot x^{4} = x^{7}$
Divide the same base: subtract them. $a^{m} \div a^{n} = a^{m - n}$	$y^{8} \div y^{3} = y^{5}$
A power of a power: multiply them. ${(a^{m})}^{n} = a^{m n}$	${(2^{3})}^{2} = 2^{6} = 64$
Anything (except 0) to the 0 is 1.	$7^{0} = 1$
A negative exponent flips it under 1. $a^{- n} = \frac{1}{a^{n}}$	$5^{- 2} = \frac{1}{25}$
A fraction exponent is a root. $a^{1 / n}$ is the nth root; $a^{m / n}$ is that root to the m.	$8^{2 / 3} = {(\sqrt[3]{8})}^{2} = 2^{2} = 4$
A negative base: an even power is positive, an odd power negative. Watch the parentheses.	${(- 3)}^{2} = 9$ , but $- 3^{2} = - 9$

Simplifying a square root

Pull out the biggest perfect square that divides it: $\sqrt{72} = \sqrt{36 \times 2} = 6 \sqrt{2}$ . Roots multiply and divide freely, $\sqrt{a} \cdot \sqrt{b} = \sqrt{a b}$ , but they do not add: $\sqrt{9} + \sqrt{16} = 7$ , while $\sqrt{25} = 5$ .

Estimating a root

Find the squares on either side: $\sqrt{50}$ is between $\sqrt{49} = 7$ and $\sqrt{64} = 8$ , and much closer to 7 (it is about 7.07).

Scientific notation

Move the decimal point until one digit is left in front, and count the moves: $45,000 = 4.5 \times 10^{4}$ , and $0.0032 = 3.2 \times 10^{- 3}$ . To multiply, multiply the fronts and add the exponents.

AFOQT questions on this: Exponent rules · Negative exponents · Fraction exponents · Simplifying roots · Scientific notation

Study guide

Exponents and roots

Simplifying a square root

Estimating a root

Scientific notation