Simplifying & manipulating expressions

Simplify.

a × a × a × a × a × a × a

4b × 5b²

Solve.

6 k - 3 = 177

Simplify.

4 x - 3 y + x + 7 y

Multiply out.

5 (2 - x)

2 x (x + 3)

Multiply out and simplify.

You must show each step in your working.

Write this expression in its simplest factorised form.

You must show your working.

Use your answer to part (a) to write down the value of x which gives the smallest value of this expression.

Simplify. 3x^2–3 × 9x⁸

Simplify.

8 j + 4 k - 10 j + 7 k

Factorise fully.

6x² + 9x

x² + 8x + 15

Here is a question and an incorrect answer.

Explain why the answer is not correct.

Show that (x + 3)(x - 5) = x² - 2x - 15.

The diagram shows a sketch of the graph y = (x + 3)(x - 5).

Complete the diagram by adding the values of the three intercepts with the axes.

The minimum point on the graph is marked T.

Write down the coordinates of the point T.

Factorise fully.

6x² + 9x

x² + 8x + 15

Here is a question and an incorrect answer.

Explain why the answer is not correct.

A sequence has nth term 2n² + 1.

Prove algebraically that the sum of any two consecutive terms in this sequence is always a

multiple of 4.

Simplify.

3 y + 6 x - y + 5 x

5 a × 6 b

In this question, all lengths are in centimetres.

The area of the rectangle is 70 cm².

Show that x² + 5x - 84 = 0.

Solve by factorising.

(b)(i) x =........ or x =........

Find the length of the longer side of the rectangle.

Simplify.

h × h × h × h

Factorise fully.

3 fg + 12 g

Expand and simplify.

(x + 3)(4 x + 1)(x - 2)

Eight consecutive numbers are written in ascending order in this grid, starting from the top and working left to right.

Kareem writes the numbers 5 to 12 in the grid.

Show that for Kareem’s grid, the sum of the numbers in the top half of the grid is 16 less

than the sum of the numbers in the bottom half of the grid.

Use algebra to prove that for any set of eight consecutive numbers written in this grid in the same way, the sum of the numbers in the top half of the grid is 16 less than the sum of the

numbers in the bottom half of the grid.

Write as a single fraction in its simplest form.

Multiply out.

5 a (2 - a)

Solve.

5 x + 9 > 12

Solve by factorising.

x² + 7x + 12 = 0

Work out.

3⁴

Simplify. 5x² x

Rearrange this formula to make x the subject.

Simplify.

4 a + 2 a + a

2 x - 3 y - 3 x + 4

Solve.

p - 5 = - 4

Factorise.

5 x - 20

Factorise fully.

14x + 7x²

The area of this rectangle can be written as ax² + bx - 10.

Find the values of a, b and c. You must show your working.

Here are two pieces of work. Each shows a question and an incorrect solution.

For each part, describe the error made and write out a correct solution.

A correct solution is........

A correct solution is........

Solve this quadratic equation by factorisation.

2x² - 6x - 24 = 5x - 3

Simplify.

3a² × 4a⁵

Factorise fully.

4x² - 12x

The area of this rectangle can be written as ax² + bx - 10.

Find the values of a, b and c. You must show your working.

Simplify fully.

x³ + 8₃x² + 15x x - 9x

The diagram shows a quadrilateral and an equilateral triangle. The perimeter of the quadrilateral is equal to the perimeter of the equilateral triangle.

Find an expression for the length of one side of the equilateral triangle. Give your answer in terms of x in its simplest form.

Multiply out and simplify.

(3 x + y)(x + 2 y)

Complete this identity by writing in the missing numbers.

4 (........ x + 1) = 14 x - 6 (x - 2) -........

Factorise x² + 10x + 24.

Write down the solutions to x² + 10x + 24 = 0.

Factorise.

9x² - 4

Solve by factorisation.

3x² - 2x - 8 = 0

Solve.

Write (2x - 5)(x + 4) in the form 2(x + a)² - b.

You must show your working.

Charlie, Dev and Eve all attempt to sketch the graph of y = (2 x - 5)(x + 4).

Charlie‛s sketch
Dev‛s sketch Eve‛s sketch

Whose sketch is the most accurate? Write down the properties of the graph that you used in making your decision.

Simplify.

5 r - 7 t - 3 r + 2 t

a × a × a × a × a

7b⁵ ÷ b

Factorise.

4 a - 12 b

Multiply out.

3 (x + 1)

3 d (d - 2)

Factorise fully 30x² + 2x - 4.

Simplify.

t + 5 t - 4 t

Factorise.

x² + 2x

Multiply out.

5 (x + 2)

Rearrange this formula to make r the subject.

p = 3 r - 5

Simplify.

2 × 3 a

Multiply out and simplify.

Factorise.

x² - 25

(x + 2)(3x + a)(bx + 3) = 6x³ + 11x² - 17x - 30

Find the value of a and the value of b.

Use algebra to prove that an odd number multiplied by a different odd number always gives an

answer that is an odd number.

You are given that

where k and m are integers.

Find the value of k and the value of m.

Multiply out and simplify.

Factorise.

x² - 25

Frankie sketches this quadratic graph.

Frankie says

The y-intercept is 15.

Show that what Frankie says could be correct.

Explain why what Frankie says may not be correct.

Simplify.

3c²d × 2d

Factorise.

35x + 7x²

Simplify.

5 t - 3 u - t + 5 u

In this question, all measurements are in centimetres.

The square and the rectangle have the same area.

Show that x² - 8x - 20 = 0.

Solve x² - 8x - 20 = 0.

Explain why one of the answers in part (b) is not possible in the context of the question.

Write down the following.

The area of the square.

The length of the rectangle.

n is a positive integer.

Prove that (2 n + 1)(n - 3)(n + 2) + 3 n (n + 7) is always even.

Multiply out and simplify.

3 (x + 2) - (x - 1)

Solve.

Factorise.

5a² - 10a

Solve by factorising.

x² + 15x + 56 = 0

Simplify.

4 a - 2 b - 2 a + 5 b

Multiply out.

4 (x + 3)

Multiply out and simplify.

(x + 5)(x - 2)

5 (2 x + 1) + c (x + d) = 12 x - 1

Work out the value of c and the value of d.

Here are two pieces of work.

For each one, describe the error made and give the complete correct solution.

Correct solution:

Correct solution:

Expand and simplify.

(x + 1)(x - 1)(x + 2)

Solve.

4 x + 3 = 13

Multiply out and simplify.

5 (2 x + 3) + 2 (x - 4)

Simplify.

Write as a single fraction in its simplest form.

Simplify.

4 a + 5 a - 7 a

3 g - 2 f + 8 g + 5 f

Factorise fully.

6 + 9 y

2x² + 6x

Multiply out.

3 (x - 2)

2 a (a + b)

Multiply out and simplify.

(4 x + y)(x - 3 y)

Martina has answered some questions on algebra. In each question, she has made an error.

Describe her error and give the correct answer to each problem.

Question 1 Simplify. 2a × a × a

Martina’s answer 4a

Martina’s error is........

Martina’s error is........

s = 225

Martina’s error is........

Nine consecutive numbers are written on a 3-by-3 grid. They are arranged, in ascending order, in a spiral as shown.

Karen writes the numbers 3 to 11 on her grid.

The total of the first column is 3 + 10 + 9 = 22.

Karen says

The total of the first column is one less than the total of the second column.

Show that this is correct for Karen’s grid.

Victor says

If any nine consecutive numbers are arranged in ascending order in this spiral on a 3-by-3 grid, the total of the first column will always be one less than the total of the second column.

Prove that Victor is correct.

Multiply out.

4 (3 x + 2)

Factorise.

3 c - 6 d

Simplify fully.

3 t + 5 u - 2 t + 3 u

6a × 2a²

Make x the subject of the formula y = x² - 1.

Simplify.

h³ × h^-3

The length of each side of a plastic cube is 2 a millimetres. The cube has mass 32a² grams.

Find an expression for the density of the cube in its simplest form. Give the units of your answer.

Simplify fully.

Solve.

Find the value of x in each of the following.

a⁴ × a³ = a^x

(b⁴)³ = b^x

Factorise fully.

18x² + 9x

Multiply out.

Give your answer in its simplest form.

32x + d + cx + 5 = 10x + 17

Work out the value of c and the value of d.

Solve by factorising.

Show the inequality x > 3 on this number line.

Simplify.

4 a + 3 c + 7 a - 5 c

Solve.

The diagram shows a square with four identical corners shaded.

The length of each side of the square is 3 x cm. The length of each shaded corner is x cm.

Show all your working.

Tenzin is given this question.

Here is his answer.

Explain why Tenzin’s answer is not correct.

Prove that 2x + 13x + 2 + x3x + 5 + 2 is a perfect square.

Gemma says The equation (2x + 1)(3x + 2) + x (3x + 5) + 2 = - 12 has no solutions. Explain Gemma’s reasoning.

Simplify.

5 x – 6 y – x + 3 y

w⁸ ÷ w²

5c²d × 3c

Work out the value of

4 x – 7 when x = 5,

Multiply out.

4 c (d – 5)

Multiply out and simplify.

(3 x + 2)(x – 4)

Solve.

3 x - 2 ⩽ 22

In this question, assume all dimensions are in centimetres.

Jess joins three tiles together to make a larger rectangle, as shown.

Write an expression for the perimeter of her rectangle. Give your answer in its simplest form.

An expression for the area of her rectangle is 6ab + 6b².

Factorise this expression fully.

Pete joins some tiles together to make a different rectangle. The area of his rectangle is 8ab + 8b².

Draw a possible arrangement of tiles for Pete’s rectangle. Write down expressions for the length and for the width of the rectangle.

Solve by factorisation.

3x² + 11x – 20 = 0

n is a positive integer.

Prove that 13 n + 3 + (3 n - 5)(2 n + 3) is a multiple of 6.

(x + a)(x + 3)(2x + 1) = bx³ + cx² + dx - 12

Find the value of a, b, c and d.

Simplify.

2 p + 5 p - 3 p

6 j + 3 k - j - 5 k

Find the value of 10 h + 6 t when h = 12 and t = 4.

Rearrange this formula to make d the subject.

e = f - 7 d

Simplify fully.

4 (c + 2 d) + 3 (3 c - 5 d)

4 a × 5 b

Factorise fully.

6 g + 8 h

5x² - 15x

Factorise.

x² – 43²

Calculate.

57² – 43²

Simplify.

a⁶ ÷ a²

(b⁵)³

Factorise.

Solve.

Expand and simplify.

(2 x - 1)(x + 5)(3 x - 2)

Factorise.

x² – 43²

Calculate.

57² – 43²

Prove that the difference between two consecutive square numbers is always odd.

The diagram shows a cross placed on a number grid.

L is the product of the left and right numbers of the cross. T is the product of the top and bottom numbers of the cross. M is the middle number of the cross.

Show that when M = 35, L – T = 99.

Prove that, for any position of the cross on the number grid above, L – T = 99.

Simplify.

7 t - 6 u + 5 t - 4 u

Factorise.

5 v + 20 w

Solve by factorising.

Multiply out.

3 x (x + 2 y)

Solve.

7 x = 28

Rearrange the equation to make x the subject.

y = 7 x - 3

Factorise.

x² - xy

x² + 8x + 12

Expand and simplify.

Factorise fully.

Simplify.

Rearrange the equation to make x the subject.

y = 7 x - 3

Factorise.

x² - xy

x² + 8x + 12

Simplify.

x + 3x - 4x + 5 is identical to x³ + ax² - 17x + b.

Find the value of a and the value of b.

Sketch the graph of y = x - 2² - 3. Show the coordinates of any turning points.

The sketch shows part of a graph which has equation y = ax² + bx + c.

Find the values of a, b and c.

Simplify.

Write as a single fraction in its simplest form.