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).

A numerical coefficient is a constant multiplier of the variables in a term.

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”.

The laws of indices - division

Maths

Converting any recurring decimal to a fraction

Checking and securing understanding of prime factorisation

Checking and securing understanding of LCM and HCF

Checking and securing understanding of roots and integer indices

The laws of indices - multiplication

The laws of indices - raising a power to a power

The laws of indices - negative and zero exponents

The laws of indices - fractional exponents

Advanced problem solving with the laws of indices

In this unit, pupils formalised their understanding of the index laws. This understanding will allow them to perform calculations with values written in standard form more efficiently.

The conditions for standard form mean that pupils have explored how different powers of 10 can be written in the prior unit. In this unit, this understanding is generalised and the index laws formalised.

In this unit pupils learn how to manipulate indices and perform calculations without conversion by understanding index laws.

Arithmetic procedures: index laws

What is the value of $$b$$ for $$b^9 \div b^4 = 6^5$$?

What is the value of $$a$$ for $$9^{20} \div 9^7 = 9^a$$?

What is the value of $$a$$ for $$0.4^8 \div 0.4^a = 0.4$$?

What is the value of $$a$$ for $$\frac{7^a}{7^7} = 7^{12}$$?

What is the value of $$a$$ for $$\frac{5^9}{5^3} = 5^a$$?

What is the value of $$a$$ for $$\frac{x^a}{x^8} = x^{10}$$?

What is the value of $$a$$ for $$a^3 \times a^7 = 12^{10}$$?

What is the value of $$a$$ for $$a^a \times a^a = 9^{18}$$?

What is the value of $$a$$ for $$5^a \times 5^a = 5^{20}$$?

What is the value of $$a$$ for $$8^6 \times 8^{-a} = 8^a$$?

What is the value of $$a$$ for $$10^3 \times 10^a = 10^3$$?

What is the value of $$a$$ for $$9^a = 81$$?

Starter Quiz

What is the value of $$a$$ for $$a^3 \times a^7 = 12^{10}$$?