Squares Cubes and Index Laws for the ESAT

Squares

The square of any number is determined by multiplying that number by itself. For example, the square of $2.5$ is written as $2.5^2$. This is calculated as $2.5 \times 2.5$, which equals $6.25$.

It is a fundamental rule that the square of either a positive or a negative number is always a positive number. For instance, $(-4)^2 = -4 \times -4 = 16$. In general, the square of any value $x$ is written as $x^2$.

Square roots

The square root of a number is defined as the positive number that, when multiplied by itself, results in the original number. The square root of $9$ is written as $\sqrt{9}$, so $\sqrt{9} = 3$.

By mathematical definition, the square root of a number is positive. However, it is possible to discuss a 'negative square root' to refer to the negative number that, when multiplied by itself, yields the original number. Thus, while the square root of $9$ is strictly $3$, the negative square root of $9$ is $-3$.

Cube numbers

The cube of a number is found by multiplying the number by itself, and then multiplying the result by the number once more. For example, the cube of $1.2$ is $1.2^3 = 1.2 \times 1.2 \times 1.2 = 1.728$.

Unlike squares, the sign of a cube matches the sign of the original number: the cube of a positive number is positive, and the cube of a negative number is negative. In general, the cube of $x$ is written as $x^3$.

Cube roots

The cube root of a number is the value that, when multiplied by itself and multiplied by itself again, results in the original number. The cube root of a positive number is positive, and the cube root of a negative number is negative. For example, since $4 \times 4 \times 4 = 64$, the cube root of $64$ is $4$.

Index numbers or powers

The power to which a number is raised is known as the index, or indices in plural form. For example, $2 \times 2 \times 2 \times 2 \times 2 = 2^5 = 32$. In this expression, $2^5$ is the value $32$ written in index form. To succeed in the ESAT, you must be able to convert numbers into index form and evaluate numbers already written in index form.

Index laws

When working with powers that share the same base number, you must apply the following index laws.

Multiplication

To multiply powers of the same base, you add the indices: $a^m \times a^n = a^{m+n}$

You should also note that when a product of two numbers is raised to a power, the power applies to both factors: $(ab)^n = a^n b^n$

Division

To divide powers of the same base, you subtract the indices: $a^m \div a^n = a^{m-n}$

Rules for Zero and One

There are three specific rules regarding the powers of zero and one:

1. Any non-zero number raised to the power $0$ is equal to $1$. In general, $a^0 = 1$ for all non-zero values of $a$.
2. Any number raised to the power of $1$ is simply the number itself. In general, $a^1 = a$.
3. The number $1$ raised to any power remains $1$.

Fractions

When a fraction is raised to a power, the power must be applied to both the numerator and the denominator: $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$

Negative powers

A number raised to a negative power is equivalent to $1$ divided by the number raised to the corresponding positive power: $a^{-m} = \frac{1}{a^m}$

Raising powers to a further power

To raise a power to another power, you multiply the indices together: $(a^m)^n = a^{mn}$

Fractional powers

Fractional indices represent roots. Specifically:

1. The power $\frac{1}{2}$ is the square root.
2. The power $\frac{1}{3}$ is the cube root.
3. The power $\frac{1}{4}$ is the fourth root.

The general formula for fractional powers is: $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Examples using indices

Write $243$ as a power of $3$ $243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^5$

Evaluate $2^7$ $2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128$

Write as a single power of $5$: $5^3 \times 5^4$ By adding the powers: $5^{3+4} = 5^7$

Write as a single power of $2$: $2^8 \div 2^5$ By subtracting the powers: $2^{8-5} = 2^3$

Evaluate $(1.2)^0 + 5^3 + 1^5 - 7^1$ Evaluating each term individually: $1 + 125 + 1 - 7 = 120$

Work out $(\frac{3}{4})^4$ Raise both parts of the fraction to the power: $\frac{3^4}{4^4} = \frac{81}{256}$

Evaluate $5^{-3}$ Using the negative power law: $5^{-3} = \frac{1}{5^3} = \frac{1}{125}$

Work out $(2^3)^{-2}$ Multiply the powers first: $(2^3)^{-2} = 2^{3 \times -2} = 2^{-6} = \frac{1}{2^6} = \frac{1}{64}$

Evaluate $49^{-\frac{1}{2}}$ $49^{-\frac{1}{2}} = \frac{1}{49^{\frac{1}{2}}} = \frac{1}{\sqrt{49}} = \frac{1}{7}$

Evaluate $16^{-\frac{3}{4}}$ $16^{-\frac{3}{4}} = \frac{1}{16^{\frac{3}{4}}} = \frac{1}{(\sqrt[4]{16})^3} = \frac{1}{2^3} = \frac{1}{8}$