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Lesson 12 of 10

The laws of indices - fractional exponents

I can use the laws of indices with fractional exponents.

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Lesson 12 of 10

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The laws of indices - fractional exponents

I can use the laws of indices with fractional exponents.

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Lesson details

Key learning points

All of the index laws can be applied with fractional exponents.
Using a calculator can help you to form an idea about how to evaluate a power containing a fractional exponent.
You can reason that a power with a fractional exponent is equivalent to finding the square root of the base.
√a = a^(1/2)

Keywords

Reciprocal - A reciprocal is the multiplicative inverse of any non-zero number. Any non-zero number multiplied by its reciprocal is equal to 1.

Common misconception

Pupils multiply the base by the fractional exponent. e.g 25^(1/2) = 12.5

The use of a calculator allows pupils to see the index of a fraction is not to be multiplied by the base. Embedding the laws of indices where two bases are the same and each number has an index of 1/2 helps recognize the 1/2 index as a square root.

Using MWB, put the number 100 in the centre. Pupils must create as many numbers with an exponent that equate to 10. E.g 10^2, (1/100)^-1, 1000^(2/3), 4x125^(2/3), etc. Display the summary of the laws of indices to helps and give access to calculators for more support.

Teacher tip

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This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Prior knowledge starter quiz

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6 Questions

Q1.
What is the value of $$x$$ for $$(m^x)^8 = m^{32}$$

Correct Answer: 4

Q2.
What is the value of $$x$$ for $$(m^{-2})^5 = m^x$$

Correct Answer: -10

Q3.
What is the value of $$x$$ for $$(m^8)^x = m^{80}$$

Correct Answer: 10

Q4.
What is the value of $$x$$ for $$\frac{1}{16} = 2^x$$

Correct Answer: -4

Q5.
What is the value of $$x$$ for $$\frac{1}{64} = x^{-6}$$

Correct Answer: 2

Q6.
What is the value of $$a$$ for $$7^9 \times 7^a = 7^6$$?

Correct Answer: -3

Assessment exit quiz

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6 Questions

Q1.
Evaluate $$36^{\frac{1}{2}}$$, giving the positive solution where necessary.

Correct Answer: 6

Q2.
Evaluate $$1000^{\frac{1}{3}}$$, giving the positive solution where necessary.

Correct Answer: 10

Q3.
Evaluate $$27^{\frac{2}{3}}$$, giving the positive solution where necessary.

Correct Answer: 9

Q4.
Evaluate $$1000^{\frac{2}{3}}$$, giving the positive solution where necessary.

Correct Answer: 100

Q5.
Evaluate $$16^{\frac{3}{4}}$$, giving the positive solution where necessary.

Correct Answer: 8

The laws of indices - fractional exponents

The laws of indices - fractional exponents

Lesson slides

Lesson details

Key learning points

Keywords

Common misconception

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.
What is the value of $$x$$ for $$(m^x)^8 = m^{32}$$

Q2.
What is the value of $$x$$ for $$(m^{-2})^5 = m^x$$

Q3.
What is the value of $$x$$ for $$(m^8)^x = m^{80}$$

Q4.
What is the value of $$x$$ for $$\frac{1}{16} = 2^x$$

Q5.
What is the value of $$x$$ for $$\frac{1}{64} = x^{-6}$$

Q6.
What is the value of $$a$$ for $$7^9 \times 7^a = 7^6$$?

Assessment exit quiz

6 Questions

Q1.
Evaluate $$36^{\frac{1}{2}}$$, giving the positive solution where necessary.

Q2.
Evaluate $$1000^{\frac{1}{3}}$$, giving the positive solution where necessary.

Q3.
Evaluate $$27^{\frac{2}{3}}$$, giving the positive solution where necessary.

Q4.
Evaluate $$1000^{\frac{2}{3}}$$, giving the positive solution where necessary.

Q5.
Evaluate $$16^{\frac{3}{4}}$$, giving the positive solution where necessary.

Q6.
Evaluate $$16^{\frac{3}{2}}$$, giving the positive solution where necessary.

Lesson appears in

UnitMaths / Arithmetic procedures: index laws

Maths

The laws of indices - fractional exponents

The laws of indices - fractional exponents

Lesson slides

Lesson details

Key learning points

Keywords

Common misconception

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.What is the value of $$x$$ for $$(m^x)^8 = m^{32}$$

Q2.What is the value of $$x$$ for $$(m^{-2})^5 = m^x$$

Q3.What is the value of $$x$$ for $$(m^8)^x = m^{80}$$

Q4.What is the value of $$x$$ for $$\frac{1}{16} = 2^x$$

Q5.What is the value of $$x$$ for $$\frac{1}{64} = x^{-6}$$

Q6.What is the value of $$a$$ for $$7^9 \times 7^a = 7^6$$?

Assessment exit quiz

6 Questions

Q1.Evaluate $$36^{\frac{1}{2}}$$, giving the positive solution where necessary.

Q2.Evaluate $$1000^{\frac{1}{3}}$$, giving the positive solution where necessary.

Q3.Evaluate $$27^{\frac{2}{3}}$$, giving the positive solution where necessary.

Q4.Evaluate $$1000^{\frac{2}{3}}$$, giving the positive solution where necessary.

Q5.Evaluate $$16^{\frac{3}{4}}$$, giving the positive solution where necessary.

Q6.Evaluate $$16^{\frac{3}{2}}$$, giving the positive solution where necessary.

Lesson appears in

UnitMaths / Arithmetic procedures: index laws

Maths

Q1.
What is the value of $$x$$ for $$(m^x)^8 = m^{32}$$

Q2.
What is the value of $$x$$ for $$(m^{-2})^5 = m^x$$

Q3.
What is the value of $$x$$ for $$(m^8)^x = m^{80}$$

Q4.
What is the value of $$x$$ for $$\frac{1}{16} = 2^x$$

Q5.
What is the value of $$x$$ for $$\frac{1}{64} = x^{-6}$$

Q6.
What is the value of $$a$$ for $$7^9 \times 7^a = 7^6$$?

Q1.
Evaluate $$36^{\frac{1}{2}}$$, giving the positive solution where necessary.

Q2.
Evaluate $$1000^{\frac{1}{3}}$$, giving the positive solution where necessary.

Q3.
Evaluate $$27^{\frac{2}{3}}$$, giving the positive solution where necessary.

Q4.
Evaluate $$1000^{\frac{2}{3}}$$, giving the positive solution where necessary.

Q5.
Evaluate $$16^{\frac{3}{4}}$$, giving the positive solution where necessary.

Q6.
Evaluate $$16^{\frac{3}{2}}$$, giving the positive solution where necessary.