Module 6 (M6) – Number and algebra - Indices

• Ten to the power of five can be written as \(10^{5}\).
  • It means \(10 \times 10 \times 10 \times 10 \times 10 = 100000\)
• \(2^{3} = 2 \times 2 \times 2 = 8\)

Indices

The power or index represents how many times the number is to multiplied by itself.

\(2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32\)

Example

Calculate \(5^{2}\).

Answer

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

Example

Find the value of \(4^{5}\).

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

Example

Calculate the value of \(2^{3} \times 3^{2}\).

Solution

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

• \(2^{3} \times 3^{2} = 8 \times 9 =72\)

Example

Calculate \(6^{2} – 2^{4}\).

Solution

\(6^{2} – 2^{4} = 36 – 16 = 20\)

Example

Find the value of \(4^{3} \div 2^{5}\).

Solution

\(4^{3} \div 2^{5} = 64 \div 32 = 2\)

To find the value of \(4^{2} \times 4^{3}\)

It can be rewritten as \(4 \times 4 \times 4 \times 4 \times 4 = 4^{5}\)

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

When multiplying, the rule is to add the indices.

To find the value of \(5^{7} \div 5^{4}\)

It can be rewritten as \((5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5) \div 5 (\times 5 \times 5 \times 5)\)

\(78125 \div 625\)

\(= 125\)

A quicker way would be \(5^{7} \div 5^{4} = 5^{7 – 4} = 5^{3} = 125\)

When dividing, the rule is to subtract the indices.

To find the value of \((3^{3})^{2}\)

It can be rewritten as \(3^{3} \times 3^{3}\)

And using Rule 1:

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

When the power is raised to another power, the rule is to multiply the indices.

To find the value of \(6^{2} \div 6^{2}\)

By using Rule 2:

\(6^{2} \div 6^{2} = 6^{2 – 2} = 6^{0} = 1\) since \(6^{2} \div 6^{2} = 1\) and \(36 \div 36 = 1\)

• \(9^{0} = 1\)

• \(m^{0} = 1\)

• \(725^{0} = 1\)

Anything to the power of zero is 1.

Example (with rule 1)

Simplify \(7^{7} \times 7^{4}\) giving the answer in index form.

Solution

\(7^{7} \times 7^{4} = 7^{7+4} = 7^{11}\)

Example (with rule 2)

Evaluate \(2^{8} \div 2^{5}\).

Solution

\(2^{8} \div 2^{5} = 2^{8–5} = 2^{3} = 8\)

Example (with rule 3)

Simplify \((a^{3})^{2}\).

Solution

\(a^{3 \times 2} = a^{6}\)

Example (with rule 4)

Evaluate \(5^{0}\).

Solution

\(5^{0} = 1\)