C1.3 Powers and roots

Learning objectives

C1.3.1Calculate 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 1 to 15, and recall of cubes and their corresponding roots of 1, 2, 3, 4, 5 and 10, For example, evaluate square and cube roots of familiar perfect powers.

A power repeats multiplication, while a matching root undoes that power. For a positive number $a$ , $a^2$ and $a^3$ are its square and cube; $\sqrt{a}$ and $\sqrt[3]{a}$ ask which numbers produce $a$ when squared or cubed.

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

$n$	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 the square table in either direction: $13^2=169$ and $\sqrt{169}=13$ . The radical $\sqrt{169}$ means the principal, non-negative square root.

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

Read the cube table in either direction: $4^3=64$ and $\sqrt[3]{64}=4$ . Cubes and cube roots also preserve sign, so $(-3)^3=-27$ and $\sqrt[3]{-27}=-3$ .

For another power or root, identify the index before calculating. For example, $5^4=5\times5\times5\times5=625$ , and $\sqrt[4]{625}=5$ because $5^4=625$ .

Expression	Safe entry and check	Result
$\sqrt{53.29}$	square-root key; check $7.3^2$	$7.3$
$\sqrt[3]{0.729}$	cube-root template; check $0.9^3$	$0.9$
$4^5-5^4$	enter each complete power before subtracting	$1024-625=399$

Do not halve a number to find its square root, and do not multiply the base by the exponent: $6^3$ is $6\times6\times6=216$ , not 18. Keep this calculation objective separate from the index laws used to simplify algebraic powers in C1.4.

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