• Ten to the power of five can be written as \(10^5\)It means \(10 \times 10 \times 10 \times 10 \times 10 = 100,000\)
• \(2^3 = 2 \times 2 \times 2 = 8\)

Examples of indices

Calculate 5²

Answer

\( 5^2 = 5 \times 5 = 25 \)

Find the value of 4⁵

Answer

\(4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 1024\)

Any calculations must follow the correct order of operations:

Brackets
Indices
Division
Multiplication
Addition
Subtraction

Calculate the value of 2³ x 3²

Following order of operation, the indices are calculated first before multiplying.

\(2^3 \times 3^2 = 8 \times 9 = \color{red} \textbf {72}\)

Examples of calculating with indices

Calculate 6² - 2⁴

Answer:\(6^2 - 2^4 = 36 – 16 = \color {red} \textbf {20}\)

Find the value of 4³ ÷ 2⁵

Answer:\(4^3 - 2^5 = 64 \div 32 = \color {red} \textbf {2}\)

Rule 1 - multiplying

To find the value of\(\color{red} \hspace{2em} 4^2 \hspace{1.45em} \times \hspace{2.75em} 4^3\)
It can be rewritten as \(\hspace{0.25em} 4 \times 4 \quad \times \quad 4 \times 4 \times 4=4^5\)

So \(4^2 \times 4^3 =4^{\color{red}{(2+3)}} =4^5 =\color{red}\textbf {1024}\)

When multiplying the rule is to add the indices.

Rule 2 - dividing

To find the value of\(\color{red} \hspace{6.35em} 5^7 \hspace{5.65em} \div \hspace{5em} 5^4 \)
It can be rewritten as \(\hspace{0.25em} 5 \times 5 \times 5\times 5\times 5\times 5\times 5\quad \div \quad \times 5 \times 5 \times 5 \times 5\)

\( = \hspace{12em}78125 \hspace{5.2em} \div \hspace{4.8em}625\)

\( =\hspace{12em} \color{red}\textbf {125}\)

A quicker way

\(5^7 \div 5^4 = 5^{(7-4)} = 5^3 = \color{red}\textbf {125} \)

When dividing the rule is to subtract the indices.

Rule 3 - raising to a power

Simplify \(\color{red} (3^3)^2 \)
It can be rewritten as \(3^3 \times 3^3 \)

And using Rule 1

\(3^3 \times 3^3 = 3^{3+3} = 3^6 = \color{red}\textbf {729}\)

When raising a power to another power multiply the indices

Rule 4 - raising to the power of 0

\(6^2 \div 6^2\)

using Rule 2

\(6^2 \div 6^2 = 6^{(2-2)} = 6^0 = \color{red}\textbf {1}\text { since } 6^2 \div 6^2 \text{ and } 36 \div 36 = 1\)

\(9^0 = 1 \hspace{5em} m^0 = 1 \hspace{5em}725^0 = 1\)

Anything to the power of zero is 1