The commutative law allows the product of two numbers represented in standard form to be easily calculated.
Using your knowledge of the index laws, you can combine powers with the same base.
Standard form calculations can be done using a calculator.

Common misconception

When multiplying terms with coefficients, pupils also add the coefficients as well as the exponents.

Pupils should be encouraged to rewrite their expression using the associative and commutative laws, with the number parts grouped and powers grouped. This hopefully avoids this error as they can see it is the product of the numbers.

Keywords

Standard form - Standard form is when a number is written in the form A × 10n, (where 1 ≤ A < 10 and n is an integer).
Exponential form - When a number is multiplied by itself multiple times, it can be written more simply in exponential form.
Commutative - The commutative law states you can write the values of a calculation in a different order without changing the calculation; the result is still the same. It applies in addition and multiplication.
Associative - The associative law states that it doesn't matter how you group or pair values (i.e. which we calculate first), the result is still the same. It applies for addition and multiplication.

Before starting on this lesson have some multiplications of numbers in standard form on the board and ask the pupils to work out the answers on MWBs. The approaches by different pupils can be shared with the class which will allow for discussion about efficiency.

Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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6 Questions

Q1.

Standard form is when a number is written in the form $$A\times10^n$$. Which of the following is true for $$A$$?

$$1 < A < 10$$

Correct answer: $$1\leq A<10$$

$$1\leq A\leq10$$

$$1< A\leq10$$

Q2.

Write $$456\text{ } 000$$ in standard form.

$$45.6\times10^4$$

$$456\times10^3$$

Correct answer: $$4.56\times10^5$$

$$4.56\times10^3$$

Q3.

Write $$0.000\text{ } 023\text{ } 5$$ in standard form.

$$2.35\times10^4$$

$$2.35\times10^{-4}$$

Correct answer: $$2.35\times10^{-5}$$

$$0.235\times10^{-4}$$

Q4.

Simplify $$x^{-2}\times x^5$$.

$$x^{-7}$$

$$x^{-5}$$

$$x^{-3}$$

$$x^{2}$$

Correct answer: $$x^{3}$$

Q5.

Simplify $$x^{4}\div x^{-5}$$.

$$x^{-1}$$

$$x^{3}$$

$$x^{5}$$

Correct answer: $$x^{9}$$

Q6.

Simplify $${x^8\times x^{-3}}\over x^4$$.

$$x^9$$

$$x^7$$

$$x^2$$

Correct answer: $$x$$

$$x^{-1}$$

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6 Questions

Q1.

Standard form is when a number is written in the form $$A × 10^n$$, where $$1 ≤ A < 10$$ and $$n$$ is an .

Correct Answer: integer, whole number

Q2.

Calculate $$(4.6\times10^4)\times(2\times10^3)$$.

$$6.6\times10^7$$

Correct answer: $$9.2\times10^7$$

$$6.6\times10^{12}$$

$$9.2\times10^{12}$$

Q3.

Calculate $$(3.8\times10^{-4})\times(2\times10)$$.

$$7.6\times10^{-5}$$

$$7.6\times10^{-4}$$

Correct answer: $$7.6\times10^{-3}$$

$$5.8\times10^{-3}$$

$$5.8\times10^{-4}$$

Q4.

Aisha correctly rewrites the calculation $$(2\times10^{-4})\times23\text{ } 000\times(2\times10^2)$$ as an equivalent calculation using standard form. Which of these shows Aisha's calculation?

$$(2\times10^{-4})\times(23\times10^3)\times(2\times10^2)$$

$$(2\times10^{-4})\times(2.3\times10^3)\times(2\times10^2)$$

Correct answer: $$(2\times10^{-4})\times(2.3\times10^4)\times(2\times10^2)$$

$$(2\times10^{-4})\times(0.23\times10^5)\times(2\times10^2)$$

Q5.

Calculate $$(2\times10^{-4})\times23\text{ } 000\times(2\times10^2)$$.

$$6.3\times10^2$$

$$6.3\times10^{-2}$$

Correct answer: $$9.2\times10^2$$

$$9.2\times10^{-2}$$

Q6.

The radius of Saturn is $$6\times10^4$$ km. What is the diameter of Saturn? Give your answer in standard form.

$$3\times10^2$$ km

$$3\times10^4$$ km

$$12\times10^4$$ km

$$12\times10^8$$ km

Correct answer: $$1.2\times10^5$$ km