Use adaptive quiz-based learning to study this topic faster and more effectively.

# Operations and indices

## The four operations

The four arithmetic operations are addition, subtraction, multiplication and division.

Addition ($+$) gives the sum of two numbers.

Subtraction ($-$) gives the difference. Order matters.

$$\Tblue{10}-\Tgreen{3}=7,\qquad \Tgreen{3}-\Tblue{10}=-7 = -(\Tblue{10}-\Tgreen{3}).$$

Addition and subtraction are inverse operations. In other words, adding a number $B$ to a number $A$ and then subtracting the number $B$ from the sum will yield the original number $A$.

$$(\Tgreen{18} - \Tblue{15}) + \Tblue{15} = 3 + \Tblue{15} = \Tgreen{18}.$$
Addition and subtraction can be visualised using a number line.

Multiplication ($\times$) gives the product of two numbers.

$$\Tgreen{5} \times \Tblue{3} = \Tblue{3} \times \Tgreen{5} = 15.$$

Division ($\div$) gives the quotient of two numbers. Order matters.

$$\Tblue{24} \div \Tgreen{6} = 4,\qquad\displaystyle\Tgreen{6} \div \Tblue{24} = \frac{1}{4} = \frac{1}{\Tblue{24} \div \Tgreen{6}}.$$

It is impossible to divide by zero.

Multiplication and division are inverse operations. In other words, multiplying a number $A$ by another number $B$ and then dividing the product by the number $B$ will yield the original number $A$.

$$(\Tblue{18}\div \Tgreen{6}) \times \Tgreen{6} = 3\times \Tgreen{6} = \Tblue{18}.$$

## Introduction to indices

Indices are used when you multiply a number by itself several times.

They make the multiplication easier to read and write.

$$\Tblue{3} \times \Tblue{3} \times \Tblue{3} \times \Tblue{3} = \Tblue{3}^\Tred{4} = 81,\quad \Tblue{-2} \times \Tblue{-2} = (\Tblue{-2})^\Tred{2} = 4$$

The number being multiplied is the base ($\Tblue{3}$). The base is raised to the power of the index ($\Tred{4}$).

This process is called exponentiation.

A power of two is a square. A power of three is a cube.

Twelve square: $12 \times 12 = 12^2 = 144$

Four cubed: $4 \times 4 \times 4 = 4^3 = 64$.

Any number except for 0 raised to the power 0 is 1. A number raised to the power of 1 is itself.

$$2^\Tred{0} = 1,\quad 2^\Tred{1} = 2,\quad 2^\Tred{2} = 2\times2 = 4,\quad 2^\Tred{3} = 2\times 2\times 2 = 8.$$
The area of a square $A$ is equal to the length $a$ of its sides squared. The volume of a cube $V$ is equal to the length $a$ of its sides cubed.

## Negative indices

A number can be raised to a negative index. It is the reciprocal of the number with the positive index.

\begin{align*} \Tblue{4}^{\Tred{-1}} &= \frac{1}{\Tblue{4}^\Torange{1}} = \frac{1}{4},\\ \Tblue{3}^{\Tred{-4}} &= \frac{1}{\Tblue{3}^\Torange{4}} = \frac{1}{81},\\ (\Tblue{-2})^{\Tred{-5}} &= \frac{1}{(\Tblue{-2})^\Torange{5}} = \frac{-1}{32}. \end{align*}

## Square and cube roots

The square root of a number is a value that, when squared, equals the original number.

A positive number has two opposite square roots.

$4$ has two square roots $2$ and $-2$, because $$2^\Tred{2} = \Tblue{4},\quad (-2)^\Tred{2} = \Tblue{4}.$$

Zero has one square root. A negative number has no real square root.

$-2$ has no square root. $0$ is the only square root of $0$!

The positive square root of $a$ is written $\sqrt{a}$.

The square roots of $2$ are $\sqrt{2}$ and $-\sqrt{2}$.

The cube root of a number is the value that, when cubed, equals the original number. The cube root of $a$ is written $\sqrt[3]{a}$.

Positive and negative numbers have exactly one cube root.

$$\sqrt[\Tred{3}]{\Tblue{64}} = 4,\quad 4^\Tred{3} = \Tblue{64},\qquad \sqrt[\Tred{3}]{\Tblue{-27}} = -3,\quad (-3)^{\Tred{3}} = \Tblue{-27}.$$
The side length of a square is the positive square root of its area.

## Fractional indices

A positive number can be raised to a fractional index.

$27^{1/3}$, $100^{3/2}$ and $\pi^{-5/7}$ have fractional indices.

For a positive number $a$, $a^{1/\Torange{q}}$ is the positive number that, when raised to the power of $\Torange{q}$, is equal to a. It is called the $q$-th root of $a$. It is also written as $\sqrt[\Torange{q}]{a}$.

$$\Tblue{9}^{1/\Torange{2}} = \sqrt{\Tblue{9}} = 3,\quad 3^\Torange{2} = \Tblue{9},\qquad \Tblue{16}^{1/\Torange{4}} = \sqrt[\Torange{4}]{\Tblue{16}} = 2,\quad 2^\Torange{4} =\Tblue{16}.$$

$a^{1/2}$ is the square root $\sqrt{a}$ of $a$. $a^{1/3}$ is the cube root.

For a positive number $a$, $a^{\Tred{p}/\Torange{q}}$ is $a^{1/\Torange{q}}$ raised to the power of $\Tred{p}$.

\begin{align*} &\qquad \Tblue{27}^{\Tred{2}/\Torange{3}} = (\Tblue{27}^{1/\Torange{3}})^\Tred{2} = 3^\Tred{2} = 9,\\ &\Tblue{100}^{\Tred{-3}/\Torange{2}} = (\Tblue{100}^{1/\Torange{2}})^{\Tred{-3}} = 10^{\Tred{-3}} = 0.001. \end{align*}

## Laws of indices

There are three laws of indices. They relate operations on exponents with the same base to operations on the indices.

• Addition of indices: When the exponents are multiplied, the powers are added. $$\Tblue{3}^\Tred{2} \times \Tblue{3}^\Torange{5} = (\Tblue{3} \times \Tblue{3}) \times (\Tblue{3} \times \Tblue{3} \times \Tblue{3} \times \Tblue{3} \times \Tblue{3}) = \Tblue{3}^7 = \Tblue{3}^{\Tred{2}+\Torange{5}}$$
• Subtraction of indices: When the exponents are divided, the powers are subtracted. \begin{align*} \Tblue{2}^\Tred{4} \div \Tblue{2}^\Torange{2} &= (\Tblue{2} \times \Tblue{2} \times \Tblue{2} \times \Tblue{2}) \div (\Tblue{2} \times \Tblue{2}) \\&= \frac{\Tblue{2} \times \Tblue{2} \times \Tblue{2} \times \Tblue{2}}{\Tblue{2} \times \Tblue{2}} = \Tblue{2}^2 = \Tblue{2}^{\Tred{4}-\Torange{2}} \end{align*}
• Multiplication of indices: When an exponent is raised to a power, the powers are multiplied. $$(\Tblue{3}^\Tred{2})^\Torange{3} = (\Tblue{3} \times \Tblue{3}) \times (\Tblue{3} \times \Tblue{3}) \times (\Tblue{3} \times \Tblue{3}) = \Tblue{3}^6 = \Tblue{3}^{\Tred{2} \times\Torange{3}}$$
Index Exponent Formula Example
$+$ $\times$ $\Tblue{a}^{\Tred{x}+\Torange{y}}=\Tblue{a}^\Tred{x}\Tblue{a}^\Torange{y}$ $\Tblue{2}^{\Tred{2}+\Torange{1}} = \Tblue{2}^\Tred{2}\cdot \Tblue{2}^\Torange{1} = \Tblue{2}^3$
$-$ $\div$ $\displaystyle \Tblue{a}^{\Tred{x}-\Torange{y}}=\frac{\Tblue{a}^\Tred{x}}{\Tblue{a}^\Torange{y}}$ $\displaystyle \Tblue{2}^{\Tred{5}-\Torange{2}} = \frac{\Tblue{2}^\Tred{5}}{\Tblue{2}^\Torange{2}} = \Tblue{2}^3$
$\times$ Power $(\Tblue{a}^{\Tred{x}})^{\Torange{y}}=\Tblue{a}^{\Tred{x}\Torange{y}}$ $(\Tblue{2}^\Tred{2})^\Torange{2}= \Tblue{2}^{\Tred{2}\times \Torange{2}} = \Tblue{2}^4$

## Order of operations

What is the value of this expression? $$(4-2)^3 - 5\div (2 + 3)$$

When there are several operations in an expression, we must do them in a specific order.

The rule is BIDMAS: Brackets first; then indices; division and multiplication next; addition and subtraction last.

$\Torange{5 \times 3} - 2 = \Tred{15 -2} = 13$ : multiplication then subtraction.

\begin{align*} \Torange{(3+4)}^2 &+ 3 \times 7 \\ &= \Tred{7^2} + 3\times 7= 49 + \Tgreen{3\times 7} = \Tblue{49 + 21} = 70.\\ \Torange{(4-2)}^3 &- 5\div \Torange{(2 + 3)}\\ & = \Tred{2^3} - 5\div 5 = 8 - \Tgreen{5\div 5} = \Tblue{8 - 1} = 7. \end{align*}

When we work through addition and subtraction, we work from left to right. For example, we work out $2-3+5$ as $$\Tred{2-3} +5 = \Tred{-1} + 5 = 4,$$ not as $2 - (\Tred{3+5}) = 2 - \Tred{8} = -6$.

The same applies to multiplication and division.