Operations and indices
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 $$A$$.
$$$ (\Tgreen{18} - \Tblue{15}) + \Tblue{15} = 3 + \Tblue{15} = \Tgreen{18}.$$$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 $$A$$.
$$$ (\Tblue{18}\div \Tgreen{6}) \times \Tgreen{6} = 3\times \Tgreen{6} = \Tblue{18}. $$$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.
$$$ 2^\Tred{0} = 1,\quad 2^\Tred{1} = 2,\quad 2^\Tred{2} = 2\times2 = 4,\quad 2^\Tred{3} = 2\times 2\times 2 = 8. $$$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*}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.
$$$ \sqrt[\Tred{3}]{\Tblue{64}} = 4,\quad 4^\Tred{3} = \Tblue{64},\qquad \sqrt[\Tred{3}]{\Tblue{-27}} = -3,\quad (-3)^{\Tred{3}} = \Tblue{-27}.$$$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*}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 | $$(\Tblue{a}^{\Tred{x}})^{\Torange{y}}=\Tblue{a}^{\Tred{x}\Torange{y}}$$ | $$(\Tblue{2}^\Tred{2})^\Torange{2}= \Tblue{2}^{\Tred{2}\times \Torange{2}} = \Tblue{2}^4$$ |
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.