KS3 Maths · Year 7 · Chapter 10: Algebra · Lesson 5

Using index notation

Year 7About 70 minutesNo calculator47 marks

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Quick questions to start.

A1
Last lesson
Write $10 x + 25$ in the form $5 (\dots\dots)$ .
[1 mark]
Answer $5 (\dots\dots)$
A2
Needed today
Work out $4^{3}$ .
[1 mark]
Answer
A3
Needed today
Write $7 \times 7 \times 7 \times 7 \times 7$ using a power.
[1 mark]
Answer
A4
From chapter 6
A cube has edges of $3$ cm. Work out its volume.
[1 mark]
Answer $\dots\dots cm^{3}$

Example, then your turn

Year 7

Your teacher works through each example with you. Then try the one beside it.

Example 1

Write each one as simply as possible, using powers.

(a) $t \times t \times t \times t \times t$ (b) $3 k \times 5 k \times k$

Hint Multiply the numbers. Then count how many times each letter is multiplied.

B1
Write each one as simply as possible, using powers.
(a) $5 w \times 2 w$
(b) $y \times 3 y \times 2 y$
(c) $2 p \times q \times 3 p$
[3 marks]
Answer(a) $\dots$ (b) $\dots$ (c) $\dots$

Example 2

Write each one as simply as possible.

(a) $(2 n)^{3}$ (b) $(5 x)^{2}$ (c) $2 x \times (3 x)^{2}$

Hint

(2 n)^{3}

means

2 n \times 2 n \times 2 n

. Every part of the bracket is cubed, not just the

n

B2
Write each one as simply as possible.
(a) $(3 t)^{2}$
(b) $(2 y)^{4}$
(c) $4 a \times (2 a)^{2}$
[3 marks]
Answer(a) $\dots$ (b) $\dots$ (c) $\dots$

Example 3

Work out the value of each expression when $n = 4$ .

(a) $2 n^{2}$ (b) $(2 n)^{2}$ (c) $n^{3} + n^{2}$

Hint Work out the power first, unless there is a bracket. In

2 n^{2}

only the

n

is squared.

B3
Work out the value of each expression when $n = 5$ .
(a) $3 n^{2}$
(b) $(3 n)^{2}$
(c) $n^{4} - 10 n^{2}$
[3 marks]
Answer(a) $\dots$ (b) $\dots$ (c) $\dots$

Example 4

Expand the brackets.

(a) $x (x^{2} + 3)$ (b) $3 a (4 a - a^{2})$

Hint Multiply every term inside the bracket by the term outside.

a \times a^{2} = a^{3}

B4
Expand the brackets.
(a) $y (y^{2} - 5)$
(b) $4 p (p + 2 p^{2})$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Practice

Year 7

C1
(a) Explain the difference between $2 k$ and $k^{2}$ .
(b) $k$ is greater than $0$ . Find the value of $k$ for which $2 k$ and $k^{2}$ are equal.
[2 marks]
Answer(b) $k = \dots\dots$
C2
Which two of these are equal to $2 x^{3}$ ?
$A x^{3} + x^{3}$ $B (2 x)^{3}$
$C 2 x \times x^{2}$ $D x^{6}$
[2 marks]
Answer $\dots\dots$ and $\dots\dots$

C3
Fill in each gap.
(a) $3 x^{2} \times □ = 12 x^{5}$
(b) $□ \times 2 y = 10 y^{3}$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$
C4
This square has sides of $3 x$ cm. Write expressions for (a) its area, (b) its perimeter.
[2 marks]
Not drawn accurately
Answer(a) $\dots\dots$ (b) $\dots\dots$
C5
Work out the value of each expression when $a = 2$ and $b = 3$ .
(a) $a^{3} b^{2}$ (b) $(ab)^{2}$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Exam-style questions

GCSE Foundation

D1
Ravi says, " $(2 x)^{3} = 6 x^{3}$ ."
Ravi is wrong. Write $(2 x)^{3}$ correctly, and show with $x = 2$ that $6 x^{3}$ gives a different value.
[3 marks]

D2
This cuboid is $5$ cm long, $x$ cm high and $x$ cm deep.
Not drawn accurately
(a)
Write an expression for its volume.
[1 mark]
Answer $\dots\dots cm^{3}$
(b)
Write an expression for its total surface area.
[3 marks]
Answer $\dots\dots cm^{2}$
(c)
Write your answer to part (b) in the form $2 x (\dots\dots)$ .
[1 mark]
Answer $2 x (\dots\dots)$
Total for D2: 5 marks
D3
Work out the value of each expression when $n = - 2$ .
(a)
$n^{3}$
[1 mark]
Answer
(b)
$(3 n)^{2}$
[1 mark]
Answer
(c)
$2 n^{2} - n^{3}$
[2 marks]
Answer
Total for D3: 4 marks
D4
Expand and simplify $x (x^{2} + 4) + 2 x (x^{2} - 1)$ .
[3 marks]
Answer $\dots\dots$
D5
Show that $(3 n)^{2} - 3 n^{2} = 6 n^{2}$ .
[3 marks]

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$n$ is a whole number from $1$ to $10$ .
For which values of $n$ is $n^{2}$ bigger than $2^{n}$ ? For which values are they equal?
[4 marks]
Hint 1 · What to try
Make a table with columns $n$ , $n^{2}$ and $2^{n}$ .
Hint 2 · The first line
$2^{n}$ doubles each time $n$ goes up by $1$ . $n^{2}$ grows more slowly once $n$ is $5$ or more.
Hint 3 · The full method
$n = 2$ : $4$ and $4$ . $n = 3$ : $9$ and $8$ . $n = 4$ : $16$ and $16$ . $n = 5$ : $25$ and $32$ .

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write a product such as $3 k \times 2 k$ using a power.
I can tell $(2 n)^{2}$ from $2 n^{2}$ , and work out their values.
I can expand a bracket that gives powers.
I can write expressions for areas and volumes using powers.

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