The laws of indices - raising a power to a power | Maths

Starter quiz

What is the value of $b$ for $b^7 \div b^4 =9^3$ ?

'9' ✓
What is the value of $c$ for $8^9 \times 8^4 = 8^c$ ?

'13' ✓
What is the value of $b$ for $7^{-9} \div 7^4 = 7^b$ ?

'-13' ✓
What is the value of $k$ for $0.6^5 \div 0.6^k = 0.6^{-1}$ ?

'6' ✓
What is the value of $k$ for $9^{(2k)} \div 9^k = 9^8$ ?

'8' ✓
What is the value of $m$ for $m^9 \div m^7 = 6^2$ ?

'6' ✓

Exit quiz

What is the value of $x$ for $(m^3)^2 = m^x$ ?

'6' ✓
What is the value of $x$ for $(p^8)^2 = p^x$ ?

'16' ✓
What is the value of $x$ for $(m^x)^6 = m^{24}$ ?

'4' ✓
What is the positive value of $x$ for $(m^x)^x = m^{16}$ ?

'4' ✓
What is the value of $x$ for $(x^3)^3 = 5^9$ ?

'5' ✓
What is the value of $x$ for $(x^x)^2 = x^6$ ?

'3' ✓

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Lesson Details

Key learning points

The index tells you how many identical terms you must multiply together.
By studying the structure of multiplication, you can see how the index will change.
When multiplying powers with the same bases, the indices can be summed.
This process can be made quicker as repeated addition can be more efficiently calculated through multiplication.
(a^b)^c = a^(bc)

Common misconception

Forgetting to raise the coefficient to the power as well as the power term.

Highlight that everything in the bracket is being affected by the power. By showing examples where the bracket is written in expanded form and then the repeated multiplication helps with this.

Keywords

Index - An exponent is a number positioned above and to the right of a base value. It indicates repeated multiplication. An alternative word for this is index (plural indices).
Coefficient - A numerical coefficient is a constant multiplier of the variables in a term.
Power - 16 is the fourth power of 2. Alternatively this can be written as 2^4 which is read as “2 to the power of 4”.