Laws of Indices

Prerequisite Materials

Surds

Describing

In index notation, the number being multiplied by itself is called the base. The number written above and to the right of the base is called the index or power or order (a superscript). For example: if the base is 3 and the index is 4 then

3 \times 3 \times 3 \times 3 = 3^{4}

Indices is the plural of index

Some powers of 10 have a name called a prefix.

The Laws

a^{n} = \overset{\overset{n}{{}}{a \times a \times \dots \times a \times a}

where there are $n$ factors of $a$ and $n \in ℕ$

$a^{1} = a$

a^{0} = 1

where $a \neq 0$

a^{-1} = \frac{1}{a}

where $a \neq 0$

a^{- n} = \frac{1}{a^{n}}

where $n > 0$

a^{½} = \sqrt{a}

where $a \geq 0$

$a^{\frac{1}{n}} = \sqrt[n]{a}$ where $n \in ℕ$

$a^{\frac{m}{n}} = \sqrt[n]{a^{m}} = {(\sqrt[n]{a})}^{m}$ where $n \in ℕ$

${(a^{m})}^{n} = a^{m n} = {(a^{n})}^{m}$

$a^{m} a^{n} = a^{m} \times a^{n} = a^{m + n}$

$a^{m} \div a^{n} = \frac{a^{m}}{a^{n}} = a^{m - n}$

${(a b)}^{n} = a^{n} b^{n}$

${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$

Videos, Practice Questions, and Worksheets

Indices (numerical): Practice Questions; Textbook Exercise
Indices: fractional: Practice Questions; Textbook Exercise
Indices: laws of: Practice Questions; Textbook Exercise
Indices: negative: Practice Questions; Textbook Exercise

Algrbraic Indices

Algebra: indices: Practice Questions; Textbook Exercise

Algebraic Indices
(Care of revisionmaths.com)

Surds

Prerequisite Materials:

Difference
of
Two Squares

List
of
Squares & Cubes

Describing Surds:

In many cases the square root of a number can be simplified to an integer or a ratio of integers.

$\sqrt{9} = 3$

$\sqrt{\frac{25}{4}} = \frac{5}{2}$

$\sqrt{10.24} = 3.2$

Surds are square roots of a number where the answer cannot be simplified into a whole or rational number.

$\sqrt{2} \approx 1.414213562$

$\sqrt{\frac{2}{3}} = \frac{\sqrt{6}}{3}$

$\sqrt{48} = 4 \sqrt{3}$

Consider an example, $\sqrt{2} \approx 1.414213562$ , but ${1.414213562}^{2} = 1.999999999$

$1.999999999$ is approxmately 2, so $1.414213562$ is also an approximation. It is more accurate if to leave the answer as a surd, √2.

A surd is a number written exactly using a root symbol.
For Example: $\sqrt{3}$ and $\sqrt[3]{5}$ are surds, $\sqrt{4}$ and $\sqrt[3]{8}$ are not as they can both be written as 2.

Rules for Surds

\sqrt{a \times a} = a

\sqrt{a b} = \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} = \sqrt{a} \sqrt{b}

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

a \sqrt{c} + b \sqrt{c} = (a + b) \sqrt{c}

Rationalising the Denominator

\frac{a}{\sqrt{b}} = \frac{a \times \sqrt{b}}{\sqrt{b} \times \sqrt{b}} = \frac{a \sqrt{b}}{b}

\frac{c}{a \pm \sqrt{b}} = \frac{c (a \mp \sqrt{b})}{(a \pm \sqrt{b}) (a \mp \sqrt{b})}

= \frac{c (a \mp \sqrt{b})}{a^{2} - b}

Follow on Materials:

Solving equations with surds

Prerequisite Materials

Surds

Describing

The Laws

Videos, Practice Questions, and Worksheets

Algrbraic Indices

Algebraic Indices (Care of revisionmaths.com)

Surds

Prerequisite Materials:

DifferenceofTwo Squares

ListofSquares & Cubes

Describing Surds:

9=3

254=52

10.24=3.2

2≈1.414213562

23=63

48=4⁢3

Rules for Surds

Rationalising the Denominator

Further reading, videos and questions:

Follow on Materials:

Algebraic Indices
(Care of revisionmaths.com)

Difference
of
Two Squares

List
of
Squares & Cubes

$\sqrt{9} = 3$

$\sqrt{\frac{25}{4}} = \frac{5}{2}$

$\sqrt{10.24} = 3.2$

$\sqrt{2} \approx 1.414213562$

$\sqrt{\frac{2}{3}} = \frac{\sqrt{6}}{3}$

$\sqrt{48} = 4 \sqrt{3}$