Powers & roots

Write 5 × 5 × 5 × 5 × 5 × 5 as a power of 5.

Write down the value of 1¹⁰.

Work out.

4 × 4 × 4 - 3³

Calculate.

Give your answer correct to 3 significant figures.

A solid steel cube has a mass of 1.05 kg. The density of the steel is 7.9 g/cm³.

Orla says,

The length of one side of the cube is approximately 10 cm.

Use estimation to decide if Orla’s statement is reasonable. Show how you decide.

Orla’s statement is........ because........

Find the value of:

2⁵,

Simplify.

y¹² ÷ y⁴

Find the value of p.

Ben thinks of a number.

Ben says,

When I square root my number and divide the result by 10 the answer is 1.3.

Find Ben’s number.

Here are the first three tile patterns of a sequence.

Pattern 1 Pattern 2
Pattern 3
Pattern 4

Draw Pattern 4 in the space above.

Complete this table.

Each pattern in the sequence can be split into a square of tiles and a single column of tiles.

For example, Pattern 3:

The square in Pattern n contains 4096 tiles.

Work out how many tiles are in Pattern n.

Write down an example of each of the following.

An even number.

A multiple of 7.

A cube number between 20 and 220.

A prime number less than 10.

Work out.

₃ 19.5⁴ - 18²

Write your answer correct to 4 significant figures.

Work out.

-7 - 4

- 42 ÷ 6

(-4)²

Use your calculator to work out.

6⁴

Complete this statement to make it correct. Give your answer as a number in ordinary form.

Work out.

3⁴

Simplify. 5x² x

Rearrange this formula to make x the subject.

Work out.

Work out.

3⁵

Find the value of y.

384 = 6 × 4^y

Write 3^–1 as a fraction.

Calculate.

Give your answer correct to 3 significant figures.

Work out.

Calculate.

Give your answer correct to 3 significant figures.

Work out.

where is a fraction in its lowest terms. Find the value of p and the value of q.

The kth term of a sequence is r^k, where r ≠ 0. The sixth term is equal to three times the second term.

Find the value of r, giving your answer correct to 3 decimal places.

Write down each of the following.

An even number between 11 and 17.

A square number between 15 and 35.

The cube root of 64.

3 is a factor of 51.

Find a factor of 51 between 10 and 20.

Use one of these symbols <, = or > to make each statement true.

4²........ 9

A student says that they have placed the following values in order starting with the smallest.

Has the student done this correctly? Show how you decide.

A student says that they have placed the following values in order starting with the smallest.

Has the student done this correctly? Show how you decide.

Simplify.

Work out.

Work out, using your calculator.

The diagram shows a triangular prism and a cube. The ends of the prism are right-angled triangles with base 16 cm and height 12 cm. The prism is 18 cm long.

The volume of the prism is equal to the volume of the cube.

Find the surface area of the cube. You must show your working.

Complete this statement by writing the missing power in the box.

The diagram shows a triangular prism and a cube. The ends of the prism are right-angled triangles with base 16 cm and height 12 cm. The prism is 18 cm long.

The volume of the prism is equal to the volume of the cube.

Find the surface area of the cube. Y ou must show your working.

A student thinks of a number. They square it and then add 6. Their answer is 295.

What number is the student thinking of?

The diagram shows a prism.

Draw an accurate plan view of the prism on the one-centimetre square grid below.

Show that the volume of the prism is 72 cm³.

A cuboid with a square base also has a volume of 72 cm³. The height of the cuboid is 2 cm.

Work out the length of one side of the square base.

Use your calculator to work out.

4⁵

Calculate.

(6² + 5)³

Write your answer correct to 3 significant figures.

Nadia thinks of a number. She finds the square root and then divides by 5. Her answer is 20.

What number is she thinking of?

Point A is plotted at (4, - 2) on this one-centimetre square grid.

Point A is a corner of a square with area 36 cm². The other corners of the square have integer coordinates and lie on the grid.

Find the coordinates for the corner of the square that is diagonally opposite point A. You may use the grid above to help you.

Complete the power of 2 for each statement by writing the missing value in the box.

2 × 2^y = 1.

Find the value of y.

A cuboid measures 6 cm by 8 cm by 15 cm. A cube has the same volume as the cuboid.

Find the surface area of the cube, giving your answer correct to 3 significant figures.

Work out.

Simplify.

The volume of a cube is 125 cm³.

Calculate the total surface area of the cube. Give the units of your answer.

Find the value of

2⁸.

The cube of 3 is added to the square root of 7.

Put a ring around the correct statement.

The diagram shows two rectangles, A and B.

Rectangle A has a width of 25 cm and a height of 12 cm. The width of rectangle B is three times the height of rectangle B.

The area of rectangle A is equal to the area of rectangle B.

Find the perimeter of rectangle B.

Complete the following.

Calculate.

Give your answer correct to 3 significant figures.

Evaluate.₁

Work out 7².

Write 6^–1 as a fraction.

Tia thinks of a number. She finds the square root and subtracts 4. Her answer is 1.

What number is she thinking of?

Work out 2⁴.

Find the value of n.

Patrick writes down a number.

He says

If I find the square root of that number and then add 15, I get 27.

What number did Patrick write down?

Work out.

10³

9(8 – 3 × 2)

Put brackets into this sum so that the answer is correct.

1 + 2 × 3 + 5 = 17

Write down.

3091 rounded to the nearest hundred

3% as a decimal

the cube root of 27

Complete the statement below using a number from this list.

- 2 0 - 6 6

Write the following numbers in order of size, smallest first.

0.4 0.5 0.06 0.444 0.46

smallest

Calculate.

Work out.

1 + 4 ÷ 2

2 + 5 × (8 - 4)

Evaluate.

2⁵

Estimate the value of

Andrew is thinking of a number.

• It is between 1 and 150.

• It is one more than a square number.

• It is three less than a cube number.

• It is not a prime number.

What is Andrew’s number? You must show all your reasoning.

Write 7 × 7 × 7 × 7 as a power of 7.

Complete this working to write 4³ as a power of 2.

4³ = 4 × 4........

so 4³ = 2 × 2 × 2 ×........

Write these numbers in order, starting with the largest.

8.1 × 10¹ 1.02 × 10³ 9.83 × 10^–2 3 × 10²

largest

Andrew is thinking of a number.

• It is between 1 and 150.

• It is one more than a square number.

• It is three less than a cube number.

• It is not a prime number.

What is Andrew’s number? You must show all your reasoning.

Work out.

Evaluate.

4^-2

Work out.

Fill in the power.

5 = 125

Find the value of y.

Find the values of z.

(b) z =........ or z =........

Calculate.

Simplify fully.

There are two errors in Sam’s method for finding the value of 64 shown below.

Find the cube root of 64 and then multiply by 2. The cube root of 64 is 4 and then 4 x 2 = 8. The negative power makes the answer negative so answer equals –8. Describe these errors and then give the correct value of 64