Numerical powers, roots, and scientific notation

This section is the numerical version: use powers and roots to calculate with known numbers. The symbolic version—exponent laws, rational exponents, and algebraic radicals—belongs in Exponents, radicals, and powers in Algebra.

Exponents

An exponent indicates repeated multiplication of a known number.

For example, five factors of $2$ can be written compactly as:

2^5 = 2\cdot2\cdot2\cdot2\cdot2 = 32

Roots

To find a numerical square root, look for the nonnegative number whose square is the radicand. For example:

\sqrt{49}=7

A numerical cube root reverses cubing for a perfect cube. For example:

\sqrt[3]{27}=3

Squares and square roots

Common perfect squares:

$n$	$n^2$
$1$	$1$
$2$	$4$
$3$	$9$
$4$	$16$
$5$	$25$
$6$	$36$
$7$	$49$
$8$	$64$
$9$	$81$
$10$	$100$
$11$	$121$
$12$	$144$

Scientific notation

Scientific notation writes a number in the form:

a \times 10^n

where:

1 \le |a| < 10

and $n$ is an integer.

Examples:

45000 = 4.5 \times 10^4

0.0032 = 3.2 \times 10^{-3}

Computing with scientific notation

For numerical multiplication or division, combine the coefficients and then adjust the powers of ten. For example:

(3\times10^4)(2\times10^5)=6\times10^9

The general exponent laws used to justify this procedure are developed in Algebra.

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