1.4 Powers and roots

Learning objectives

1.4A Square numbers and cube numbersidentify square numbers and cube numbers 1.4B Squares, square roots, cubes and cube rootscalculate squares, square roots, cubes and cube roots 1.4C Index notation and index lawsuse index notation and index laws for multiplication and division of positive and negative integer powers including zero 1.4D Prime factor decompositionexpress integers as a product of powers of prime factors 1.4E Highest common factors (HCF) and lowest commonfind highest common factors (HCF) and lowest common multiples (LCM)

A square number is the product of an integer with itself, $n^2=n\times n$ . A cube number is the product of three equal integer factors, $n^3=n\times n\times n$ .

Type	Sequence from non-negative integers	Recognition
squares	$0,1,4,9,16,25,36,\ldots$	$25=5^2$
cubes	$0,1,8,27,64,125,\ldots$	$27=3^3$

Some numbers are both: $64=8^2=4^3$ . A negative integer can be a cube, such as $-27=(-3)^3$ , but an integer square cannot be negative.

To test a positive integer, find whether its square root or cube root is an integer.

A number containing a square digit is not necessarily square. The whole number must equal $n^2$ for an integer $n$ .

Squaring and taking the principal square root are inverse operations for non-negative numbers; cubing and taking the cube root are inverse operations for all real numbers.

Power	Inverse statement
$14^2=196$	$\sqrt{196}=14$
$19^3=6859$	$\sqrt[3]{6859}=19$
$(-4)^3=-64$	$\sqrt[3]{-64}=-4$

The symbol $\sqrt{a}$ means the non-negative principal root. Thus $\sqrt{49}=7$ . But the equation $x^2=49$ has two solutions, $x=7$ and $x=-7$ .

Brackets matter: $(-3)^2=9$ , while $-3^2=-(3^2)=-9$ under the operation hierarchy.

There is no real square root of a negative number, but negative numbers do have real cube roots.

In $a^n$ , $a$ is the base and $n$ is the index. Index laws compress repeated multiplication and extend consistently to zero and negative integer powers.

Law, where defined	Result
$a^m\times a^n$	$a^{m+n}$
$a^m\div a^n$	$a^{m-n}$
$(a^m)^n$	$a^{mn}$
$a^0$ , $a\ne0$	$1$
$a^{-n}$ , $a\ne0$	$\frac1{a^n}$

$2^{-3}\times2^9=2^{-3+9}=2^6$ . Also, $5^{-2}=\frac1{25}$ ; the negative index creates a reciprocal, not a negative value.

Multiplication and division laws require the same base. For division and negative powers the base must be non-zero.

Do not multiply indices when multiplying powers: $a^m a^n=a^{m+n}$ . Indices multiply only in a power of a power.

A prime factor decomposition writes an integer greater than $1$ as a product containing only prime factors. Repeated primes are collected as powers.

Step	Example for $720$
divide repeatedly by prime numbers	$720=2\times360=2^2\times180=2^3\times90=2^4\times45$
continue until every remaining factor is prime	$45=3\times3\times5$
collect repeated factors	$720=2^4\times3^2\times5$

Multiply the prime powers to check: $16\times9\times5=720$ . A different valid factor tree must finish with the same prime factors.

The number $1$ is neither prime nor included as a prime factor; including it changes nothing but is not part of the decomposition.

Stopping at composite factors is incomplete. For example, $720=16\times45$ is a factorisation, but not yet a product of prime factors.

The highest common factor divides every given number and is as large as possible. The lowest common multiple is divisible by every given number and is as small as possible.

Number	Prime decomposition
$72$	$2^3\times3^2$
$108$	$2^2\times3^3$

For the HCF, keep only primes present in both and choose the smaller exponent: $2^2\times3^2=36$ .

For the LCM, include every prime present and choose the larger exponent: $2^3\times3^3=216$ . Check that $216$ divides exactly by both $72$ and $108$ .

Do not swap the exponent rules: HCF uses shared minimum powers; LCM uses all maximum powers.

A surd is an exact irrational root, such as $\sqrt2$ or $\sqrt[3]{5}$ , left in root form rather than replaced by a rounded decimal.

Root	Type	Exact simplified value
$\sqrt{49}$	rational	$7$
$\sqrt8$	surd	$2\sqrt2$
$\sqrt[3]{27}$	rational	$3$

Extract perfect-power factors: $\sqrt{ab}=\sqrt a\sqrt b$ for non-negative $a,b$ , so $\sqrt{72}=\sqrt{36\times2}=6\sqrt2$ .

Surd form preserves exact value. A calculator decimal such as $1.414\ldots$ approximates $\sqrt2$ and should not replace it when an exact answer is required.

Roots do not distribute over addition: $\sqrt{a+b}$ is not generally $\sqrt a+\sqrt b$ .

Simplify surds first, then combine only like surds: $3\sqrt8+\sqrt{32}=6\sqrt2+4\sqrt2=10\sqrt2$ .

Job	Exact move	Example
multiply surds	multiply coefficients and radicands	$\sqrt3\times\sqrt{12}=\sqrt{36}=6$
remove a single surd denominator	multiply top and bottom by that surd	$\frac1{\sqrt2}=\frac{\sqrt2}{2}$
remove a binomial surd denominator	multiply by its conjugate	$\frac2{2-\sqrt3}\times\frac{2+\sqrt3}{2+\sqrt3}=4+2\sqrt3$

Conjugates use difference of squares: $(a-b\sqrt c)(a+b\sqrt c)=a^2-b^2c$ , which is rational.

Expand brackets carefully and simplify every square root before collecting terms.

Unlike surds cannot be added: $\sqrt2+\sqrt3$ does not become $\sqrt5$ .

Fractional indices represent roots, while negative indices represent reciprocals. Together they extend the same index laws used for integer powers.

Form	Meaning	Example
$a^{1/n}$	$\sqrt[n]{a}$	$8^{1/3}=2$
$a^{m/n}$	$(\sqrt[n]{a})^m$	$8^{2/3}=2^2=4$
$a^{-p}$	$\frac1{a^p}$	$625^{-1/2}=\frac1{25}$

Apply the root and reciprocal meanings in either safe order: $16^{-3/4}=1/(16^{1/4})^3=1/8$ .

For real-number work, even roots require a non-negative radicand. Negative powers require a non-zero base.

A negative index does not make the value negative, and $a^{m/n}$ does not mean $a^m\div a^n$ .

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1.4 Powers and roots

Learning objectives

Recognise square and cube numbers

Calculate powers and their roots

Use integer index laws

Prime factor decomposition in index form

Find HCF and LCM from prime powers

Understand exact surds

Manipulate surds and rationalise denominators

Use fractional and negative indices

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