Simplifying and manipulating expressions

Simplify 45e

Solve f 14 29

Simplify 4e × 7f

Simplify d × d × d × d × d

Simplify 3a + 2k + a – 7k

Solve x + 6 = 15

Solve 2r – 9 = 14

Expand x x 7

Factorise 8y 10

m 5a 7b

Work out the value of a when m 48 and b 3

Show clear algebraic working.

Factorise y² – 11y + 30

Hence solve y² – 11y + 30 = 0

Expand x(x 3)

Simplify a⁶ a¹⁰

Simplify c³⁰ c¹²

Factorise y² 10y 21

Hence, solve y² 10y 21 0

Simplify a⁶ a¹⁰

Simplify c³⁰ c¹²

Factorise y² 10y 21

Hence, solve y² 10y 21 0

Prove that, for any three numbers which are consecutive multiples of 4, the difference between the square of the largest number and the square of the smallest number is always a multiple of 64

Show clear algebraic working.

Simplify d + d + d + d

Simplify w w w w w

Solve 6x = 42

Simplify 7r + 9y + 3r – 4y

Simplify a + a + a + a + a

Simplify b c 7

Solve 5d 40

Here is a number machine.

Find the output when the input is 10

Find the output when the input is −2

The bar chart gives information about the results of the games played by a school team.

The team gained

3 points for each game it won 1 point for each game it drew 0 points for each game it lost

The team drew 14 games.

Work out the total number of points the team gained.

Factorise 15 5x

d = 5p + 7r

Work out the value of p when d = 35 and r = 2

Sweets are sold in small packets and large packets.

There are 12 sweets in a small packet. There are 25 sweets in a large packet.

Ben buys m small packets of sweets and n large packets of sweets.

Write an expression, in terms of m and n, for the total number of packets of sweets Ben buys.

Write an expression, in terms of m and n, for the total number of sweets Ben buys.

Factorise fully 18c 45cd

Show clear algebraic working.

12Expand and simplify 3x2x 57x 4

Find the value of p

Show clear algebraic working.

Simplify 8d 4d 7d

Simplify m 10 p

Solve x 5 19

Simplify a a a a

k 8 and n = 12

Work out the value of 9k 4n

Simplify p^{3 5}

Expand and simplify 2n4n 3 nn 4

Show clear algebraic working.

Factorise x² 5x 24

Hence, solve x² 5x 24 0

Solve the inequality 3y 5 7y 10

Show clear algebraic working.

Simplify 10p – 4p + 9p

Simplify 94q

Solve 4r = 15

Simplify 10x – 7y – 6x + 4y

T = 4d – 6e

Work out the value of T when d = 13 and e = 7

Solve 5p + 11 = 28

Simplify c + c + c + c + c

Simplify 7w + 10y ‒ 9w + 2y

The nth term of a sequence is given by 7n – 4

Find the 1st term and the 5th term of the sequence.

Solve 7(x + 5) = 8 ‒ 3x Show clear algebraic working.

Kye has some toy cars.

He has n red cars. He has twice as many blue cars as red cars. He has 7 more green cars than red cars.

Write an expression, in terms of n, for the total number of red cars, blue cars and green cars that Kye has. Write your answer in its simplest form.

The total number of cars that Kye has is T

Write an expression, in terms of T and n, for the number of cars that Kye has that are not red cars, blue cars or green cars.

Expand and simplify (m + 5) (m – 8)

Show clear algebraic working.

Simplify g⁹ g²

Expand 5k²(k³ + 4)

Factorise x² 2x 63

Hence, solve x² 2x 63 = 0

Solve the inequality 7 2y 3y 12

Expand and simplify (m + 5) (m – 8)

Show clear algebraic working.

Simplify g⁹ g²

Expand 5k²(k³ + 4)

Factorise x² 2x 63

Hence, solve x² 2x 63 = 0

Solve the inequality 7 2y 3y 12

Factorise fully 6y² – 5y – 4

Express as a single fraction in its simplest form.

Simplify c × c × c

Simplify 12d × 3e

Solve 2g 3 6

Expand x(x – 4)

P 4y² w

Work out the value of P when y = −3 and w = 2

Show clear algebraic working.

Simplify (e + 12)⁰ where e > 0

Factorise fully 20x⁵y + 12x³y⁴

Factorise fully 12 pq 18p

There are 56 metal bars in a box. Each metal bar is gold or silver or zinc.

y metal bars are gold. (3 y + 7) metal bars are silver. (2 y − 5) metal bars are zinc.

Work out the number of metal bars that are zinc. Show clear algebraic working.

Here is a number machine.

When the input is 7 the output is 60

Work out the value of y

Here is a different number machine.

The input is x

Write down an expression, in terms of x, for the output.

Expand x (x + 3)

Factorise 8p + 10

Make e the subject of y = eh ‒ f

Janya thinks of a whole number. She calls her whole number w Janya writes down this information about her whole number.

Write down the possible values of w

Expand and simplify (2 x + 3)(x – 5)(x + 4)

Simplify p + p + p + p

Simplify 5e + 6f + 7e – 2f

Solve 13 – x = 7

Solve 4y + 7 = 43

Simplify 6a × 2c

Simplify 4d + 3e + d – 5e

Solve 4x – 7 = 23

Here is a number machine.

Work out the output when the input is 14

Here is a different number machine.

When the input is 11 the output is 64

Write a number on the dotted line to complete the number machine.

Simplify (2c⁴d⁷)³

Find the value of 5y⁰ where y 0

Factorise fully 16a²b³ + 20a³b

Factorise x² + 9x – 22

Hence solve x² + 9x – 22 = 0

Expand 4 x (x – 5)

Factorise y² – 9y + 20

Simplify (2c⁴d⁷)³

Find the value of 5y⁰ where y 0

Factorise fully 16a²b³ + 20a³b

Factorise x² + 9x – 22

Hence solve x² + 9x – 22 = 0

Expand 4x(x – 5)

Factorise y² – 9y + 20

14Factorise fully 50g² – 18

Expand x (3 – x)

Factorise fully 12a – 18b

There are 8 slices of cheese in each small pack of cheese. There are 20 slices of cheese in each large pack of cheese.

Afreen buys h small packs of cheese and j large packs of cheese. She buys a total of T slices of cheese.

Write down a formula for T in terms of h and j

15Finn is asked to find the value of 5 + 3² + 12

Here is his working and his answer.

5 + 3² + 12 = 8² + 12 = 64 + 12 = 76

Finn’s answer is wrong.

Explain what Finn has done wrong in his working.

Write one pair of brackets in this calculation so that the answer is correct.

Work out the value of x² + 5y when x = –3 and y = 2

Complete the table of values for y = x² – x – 4

On the grid below, draw the graph of y = x² – x – 4 for values of x from –3 to 3

–3 –2 –1 O 1 2 3 x

Expand and simplify (x + 6)(3 x – 2)(x + 6)

Simplify 10y – y

Simplify 3p4p

Solve 7x = 42

Solve n + 6 = 5

Simplify 8c + 5d – 2c – 3d

Simplify 5p × 9k

Simplify 3e + 2f + 8e – 7f

Solve 2d + 7 = 16

Solve 3(2 – 4 x) = 5 – 8 x Show clear algebraic working.

Factorise fully 18c³d² – 21c²

Factorise y² – 3y – 18

Hence, solve y² – 3y – 18 = 0

Factorise y² – 2y – 48

Write down the inequality shown on the number line

–2 –1 0 1 2 3 4 5

Solve the inequality 7w + 6 12w + 14

Factorise fully 18c³d² – 21c²

Factorise y² – 3y – 18

Hence, solve y² – 3y – 18 = 0

Factorise y² – 2y – 48

Write down the inequality shown on the number line

–2 –1 0 1 2 3 4 5

Solve the inequality 7w + 6 12w + 14

13Expand and simplify 3x(2x – 5)² Show clear algebraic working.

Complete the table of values for y = x³ – 3x + 2

On the grid, draw the graph of y = x³ – 3x + 2 for values of x from –2 to 2

By drawing a suitable straight line on the grid, use your graph to find an estimate for the solution of

Give your answer correct to one decimal place.

Simplify c × c × c c c

Solve 5 + x = 12

Expand 5(2 + 3h)

Factorise g² + 7g

Here is a number machine.

Work out the output when the input is 9

Here is a different number machine.

When the input is 30 the output is 18

Find a suitable way to complete the number machine.

The following rule is used to work out the total cost, in euros, of hiring a cement mixer.

James hires a cement mixer for 3 days.

Work out the cost to James of hiring the cement mixer.

The cost to Sophia of hiring a cement mixer is 61 euros.

For how many days does Sophia hire the cement mixer?

Expand x (10 – x)

Factorise 6y + 27

Make m the subject of the formula h = + 4

Solve 7g + 3 = 2g – 5 Show clear algebraic working.

Write down the value of (m + 2)⁰ where m is a positive integer.

Simplify (3a²b⁴)³

Factorise fully 14x²y⁴ + 21x³y²

The diagram shows a straight line drawn on a grid.

Write down an equation of the line.

Write down the value of (m + 2)⁰ where m is a positive integer.

Simplify (3a²b⁴)³

Factorise fully 14x²y⁴ + 21x³y²

The diagram shows a straight line drawn on a grid.

Write down an equation of the line.

Expand and simplify 5x(3x + 4)(2x – 1)

Simplify 12a + 3a – 7a

Simplify 8 3b

Simplify 12g − 8e − 5g + 6e

A = 3b − 5c

Work out the value of A when b = 12 and c = 4

Solve 4p + 9 = 24

Expand and simplify (n − 6)(n + 4)

Show clear algebraic working.

Expand and simplify (n − 6)(n + 4)

Show clear algebraic working.

Expand 3c³(c + 4)

Factorise x² + 8x – 9

Hence, solve x² + 8x –9 = 0

Expand and simplify (3 x – 1)(x + 2)(3 x + 1)

Simplify aa×aa×a

Simplify 8b×3c

Expand 3(x + 4)

Q = 5v² – w

Work out the value of Q when v = and w =

Show clear algebraic working.

Simplify a⁰where a 0

Factorise fully 10c³d² + 15cd⁴

6

Show clear algebraic working.

Simplify a⁰where a 0

Factorise fully 10c³d² + 15cd⁴

Solve 5x = 30

Solve y – 7 = 12

Simplify h + h + h + h + h

Simplify 5a + 7f – 2a + 4f

Simplify 3c + 5d – c + 2d

Simplify 8e 5f

Solve 5r – 3 = 8

Alisa, Jena and Mikael each pick cucumbers.

Alisa picks C cucumbers. Jena picks 5 fewer cucumbers than Alisa. Mikael picks twice as many cucumbers as Alisa.

The total number of cucumbers picked by Alisa, Jena and Mikael is T

Find a formula for T in terms of C Give your formula in its simplest form.

Solve the inequality 5x – 7 ≤ 2

Factorise y² – 2y – 35

Hence, solve y² – 2y – 35 = 0

Simplify x⁴ x⁵

Simplify (4y²)³

Factorise n² – 7n + 12

Solve the inequality 5x – 7 ≤ 2

Factorise y² – 2y – 35

Hence, solve y² – 2y – 35 = 0

Expand and simplify 5 x (x + 2)(3 x – 4)

(b) Simplify completely

Solve 5 c = 15

Expand x(8 – x)

T = 5m – 6n

Work out the value of T when m = 4.2 and n = –2.5

Make g the subject of k = 2g + t

Expand and simplify (y + 4)(2 − y)

Factorise fully 15b⁵c − 35b³c⁹

Prove that when the sum of the squares of any two consecutive odd numbers is divided by 8, the remainder is always 2 Show clear algebraic working.

Here is a rectangle.

(2 x + 3) cm
Diagram NOT

Given that the area of the rectangle is less than 75 cm²

find the range of possible values of x

Solve 5x = 20

Simplify 3a 8b

Simplify 8w – 4y + w – 3y

Factorise fully 16 + 12t

Simplify e⁸ e²

Expand and simplify (x – 3)(x + 1)

Simplify e⁸ e²

Expand and simplify (x – 3)(x + 1)

Factorise 6y² – y – 5

Express 4x² – 8x + 7 in the form a(x + b)² + c where a, b and c are integers.

Simplify 10x + 4y + 3x – 6y

Solve 2n + 5 = 16

Simplify aa aa

Simplify 4b5c

Simplify 6d + 2e + d – 5e

Expand x (4 – x)

t = ab – c

a = 1.5 b = 2.4 c = –5.6

Work out the value of t.

Make d the subject of y = dx – e

Expand 5(3 a + 4)

Factorise 4c – 14

Solve 5x – 11 = x + 6 Show clear algebraic working.

Factorise fully 15y⁴ + 20uy³

Show clear algebraic working.

Write down the value of g⁰

Factorise x² – 11x + 24

Simplify (2x³y⁵)⁴

Factorise x² + 5x – 36

Hence, solve x² + 5x – 36 = 0

Factorise fully 15y⁴ + 20uy³

Show clear algebraic working.

Write down the value of g⁰

Factorise x² – 11x + 24

Simplify (2x³y⁵)⁴

Factorise x² + 5x – 36

Hence, solve x² + 5x – 36 = 0

Give your answer in its simplest form.

Expand and simplify 2x(x – 5)(x – 3)

Expand and simplify (x + 4)(x – 2)(x + 1)

Express x² – 10x + 40 in the form (x + a)² + b, where a and b are integers.

Simplify p + p + p + p + p + p

Simplify 5y² + 6y² – 3y²

Simplify eeeee

Simplify 5c4d

Solve x – 7 = 19

18² + 15² – 5³ = 4n

Work out the value of n.

Factorise 9t – 6

Expand and simplify 4 x (2 x + 5) – 3 x (2 x – 3)

work out the value of n.

Solve the inequality 7t – 8 < 2t + 7

On the number line below, represent the solution set of the inequality solved in part (c)(i)

–5 –4 –3 –2 –1 0 1 2 3 4 5

w = 5y² – y³

Work out the value of w when y = –2

Factorise fully 8p² – 2p

Expand 4t(3t – 2)

Expand and simplify (5x – 2)(x + 4)

Simplify w + w + w + w – w

Simplify 4a2

Simplify f f fff

Simplify 4c + 4h + 5c – 6h

Factorise 10d + 15

Make t the subject of e = 7t + g

Simplify 5c d

Solve x + 5 = 12

Solve 9y = 36

Simplify 8k + 5m – 2k + 6m

Expand 4(3g + 1)

c = 4 d = 7

Work out the value of 3c + 2d

p = −6 m = −2

Work out the value of 2p² + 3m

There are 6 eggs in a small box of eggs. There are 12 eggs in a large box of eggs.

Alex buys g small boxes of eggs and h large boxes of eggs. He buys a total of T eggs.

Write down a formula for T in terms of g and h.

Simplify h⁷h²

G = c² – 4c

Find the value of G when c = –5

Show clear algebraic working.

Solve 5(4 – x) = 7 – 3 x Show clear algebraic working.

Factorise fully 16m³g³ + 24m²g⁵

Factorise y² – 2y – 48

Hence, solve y² – 2y – 48 = 0

Solve 5(4 – x) = 7 – 3x Show clear algebraic working.

Factorise fully 16m³g³ + 24m²g⁵

Factorise y² – 2y – 48

Hence, solve y² – 2y – 48 = 0

Expand and simplify (4 x + 1)(x – 3)(5 x + 6)

Expand 4(m + 2)

Solve 2x + 5 = –19

Factorise 25 f – 10

Make y the subject of the formula c = 5y – h

Solve the inequality 4x + 7 > 2

Factorise x² – x – 42

Solve the inequality 3x + 15 < 8x + 3

Show clear algebraic working.

Simplify 6m− 2k+ 5m−k

P= 2a+ 3b

Work out the value of P when a= 5 and b= 8

P= 2a+ 3b

Work out the value of a when P= 16 and b= 20

Make a the subject of d = g + 2 ac

Factorise fully 9e f− 12f

Expand and simplify (x+ 2)(x− 5)

Expand and simplify (m − 8)(m + 5)

Factorise fully 5y+ 20y²

Simplify (p²+ 3)⁰

Show clear algebraic working.

Make a the subject of d=g+ 2ac

Factorise fully 9e f− 12f

Expand and simplify (x+ 2)(x− 5)

Expand and simplify (m− 8)(m+ 5)

Factorise fully 5y+ 20y²

Simplify (p²+ 3)⁰

Show clear algebraic working.

The diagram shows a cuboid of volume Vcm³

Show that V= 15 + 16x−x²− 2x³

There is a value of x for which the volume of the cuboid is a maximum.

Find this value of x. Show your working clearly. Give your answer correct to 3 significant figures.

Simplify k + k + k + k

Write f as a single power of 9

Write f as a single power of 3

Write 5¹⁷ 5² as a single power of 5

Write 800 as a product of its prime factors. Show your working clearly.

Expand x (5 – x)

Factorise 3y – 21

Make p the subject of the formula f = 3p – d

Sergio buys m boxes of seeds and n packets of seeds.

Each box contains 10 seeds. Each packet contains 6 seeds.

The total number of seeds that Sergio buys is T.

Write down a formula for T in terms of m and n.

Expand e (3 e – 5)

Factorise 35 + 5f

Simplify (4pq²)³

N is a multiple of 5

A = N + 1 B = N – 1

Prove, using algebra, that A² – B² is always a multiple of 20

Simplify 6e 2f

Simplify 5m + 7km + k

Solve 5 y + 3 = 14

P = 2 g + 3 h

Work out the value of P when g = 7 and h

Simplify e⁹ ÷ e⁵

=Simplify (௘y²)⁸

Expand and simplify (x + 9)(x – 2)

Factorise fully 16c⁴p² + 20cp³

Complete the table of values for y = x²x

On the grid, draw the graph of y = x²x í 1 for all values of x from í2 to 4

Simplify e⁹ ÷ e⁵

Simplify (௘y²)⁸

Expand and simplify (x + 9)(x – 2)

Factorise fully 16c⁴p² + 20cp³

Complete the table of values for y = x²x

On the grid, draw the graph of y = x²x í 1 for all values of x from í2 to 4

Factorise 2x²x + 6

Show clear algebraic working.

Expand and simplify (x + 2)(2 x + 3)(x – 7) Show your working clearly.

Complete the table of values for y = 1 + 5xx²

x		0	1	2	3	4	5	6
y		1		7	7		1

On the grid, draw the graph of y = 1 + 5xx² for values of x from í1 to 6

Factorise x²x

Simplify 6x + 8xx

Simplify 4 e × 5 f

Solve 8 p = 24

Solve k

Simplify 10t + 4dt + 2d

Simplify y⁵ × y⁹

Simplify (2m³)⁴

Solve 5(x + 3) = 3x Show clear algebraic working.

Factorise x² + 2x

Hence, solve x² + 2x

Simplify y⁵y⁹

Simplify (2m³)⁴

Solve 5(x + 3) = 3 x – 4 Show clear algebraic working.

Factorise x² + 2x – 24

Hence, solve x² + 2x – 24 = 0

E = n² + n + 5

Ali thinks that the value of E will be a prime number for any whole number value of n.

Is Ali correct? You must give a reason for your answer.

Expand and simplify (2 x – 1)(x + 3)(x – 5)

Solve 3x² + 6x – 5 = 0 Show your working clearly. Give your solutions correct to 3 significant figures.

Simplify (2e² f³)³

Expand and simplify (3xy)(x + 3y)

Find the value of k.

Find the value of 25 – 4 g when g = –3

Expand and simplify x (2 x + 1) + 3(x – 2) + 7

Make a the subject of the formula M = ac – bd

Solve the inequality 5x – 4 39

Factorise fully 18e²f³ – 12e³ f

Expand and simplify 3(c – 7) + 2(3 c + 4)

Expand and simplify (x + 7)(x – 2)

Factorise fully 28y²y

Make a the subject of the formula M = ac – bd

Solve the inequality 5x – 4 39

Factorise fully 18e²f³ – 12e³ f

Expand and simplify 3(c – 7) + 2(3 c + 4)

Expand and simplify (x + 7)(x – 2)

Factorise fully 28y² – 21y

Show clear algebraic working.

Make e the subject of the formula h = 3 e + f

Complete the table of values for y = x³ – 2x² – 3x + 4

On the grid, draw the graph of y = x³ – 2x² – 3x + 4 for values of x from –2 to 3

By drawing a suitable straight line on the grid, find estimates for the solutions of the equation x³ – 2x² – x + 1 = 0 Give your solutions correct to 1 decimal place.