E1.3 Powers and roots

Learning objectives

E1.3.1Calculate with the following: • squares • square roots • cubes • cube• Calculate with the following: • squares • square roots • cubes • cube roots • other powers and roots of numbers. Includes recall of squares and their corresponding roots from

A power tells you how many times to use a number as a factor. A matching root reverses that operation: $\sqrt{a}$ asks for the non-negative number whose square is $a$ , while $\sqrt[3]{a}$ asks for the number whose cube is $a$ .

$(a^2)^{1/2}=a\quad(a\ge0),\qquad (a^3)^{1/3}=a$

$n$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
$n^2$	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225

Read every square fact in both directions: $13^2=169$ , so $\sqrt{169}=13$ . The symbol $\sqrt{169}$ means the principal square root, so its value is $13$ , not $\pm13$ .

$n$	1	2	3	4	5	10
$n^3$	1	8	27	64	125	1000

Read cube facts in both directions: $4^3=64$ , so $\sqrt[3]{64}=4$ . Cubing and cube-rooting preserve sign: $(-3)^3=-27$ and $\sqrt[3]{-27}=-3$ .

For any other power or root, match the index. For example, $5^4=5\times5\times5\times5=625$ , and $\sqrt[4]{625}=5$ because $5^4=625$ . A root with no small index shown is a square root.

Calculation	Working and check	Result
$\sqrt[4]{39\frac1{16}}$	$39\frac1{16}=39.0625$ ; check $2.5^4$	$2.5$
$\dfrac{4.87-2.7}{-0.2+\sqrt[3]{0.729}}$	$\sqrt[3]{0.729}=0.9$ , then evaluate numerator and denominator	$2.17\div0.7=3.1$
$6^3$	$6\times6\times6$	$216$

Do not multiply the base by the exponent: $6^3$ is not $18$ . Do not halve a number to find its square root. When entering a compound expression on a calculator, keep each root and its radicand grouped, and check a root by raising the result to the matching power. Index laws, negative indices and fractional-index rules are taught separately in E1.7.

Next Topic

—E1.4 Fractions, decimals and percentagesRead