Powers and roots

Eight cards with numbers on them are shown below.

Jo has four of these cards in her hand.

Dafydd chooses one of Jo’s four cards at random.

• There is an even chance that the number on the card is a square number.

• It is unlikely that the number on the card is a factor of 21.

Which four cards could Jo have in her hand?

Write the numbers on the four cards below.

Calculate 7² × 2³.

Write down the next two numbers in the following sequence.

Find the value of each of the following.

60% of 315

3⁵

Using only the numbers listed below,

120 121 122 123 124 125 126

write down

a square number,

a cube number,

a multiple of 14.

Calculate 7² × 2³.

Calculate 0∙04 × 0∙9.

Calculate 11% of 8400.

Write down the next two numbers in the following sequence.

Using only the numbers listed below,

120 121 122 123 124 125 126

write down

a square number,

a cube number,

a multiple of 14.

Megan has the following 7 playing cards.

She turned these cards face down. Megan then chose a card at random and recorded the number.

What is the probability that Megan recorded the number 5?

Circle your answer.

What is the probability that Megan recorded a square number?

Circle your answer.

Megan chooses a card at random 91 times. How many times would you expect Megan to record a square number?

You must show all your working.

Megan has the following 7 playing cards.

She turned these cards face down. Megan then chose a card at random and recorded the number.

What is the probability that Megan recorded the number 5?

Circle your answer.

What is the probability that Megan recorded a square number?

Circle your answer.

Megan chooses a card at random 91 times. How many times would you expect Megan to record a square number?

You must show all your working.

Write down the value of each of the following.

7²

Write 19·731 correct to 1 decimal place.

Write 65·4279 correct to 3 decimal places.

There is one square number between 180 and 200.

What is this square number?

A number has exactly four factors. All of the factors are less than 30. Three of the factors are 1, 2 and 26.

What is the other factor?

In this part of the question, you will be assessed on the quality of your accuracy in writing.

Which length is exactly halfway between 280 m and 410 m? You must show all your working. [2 + 1 W]

Evaluate each of the following.

9² × 10³

0·8 × 0·25

13·4 − 2·96

Give your answer as a fraction in its simplest form.

Write the reciprocal of 4 as a decimal.

You must show all your approximations in your working.

Evaluate

Give your answer in its simplest form.

What is the greatest integer value of n if 2n < 17?

What is the least integer value of n if 2ⁿ > 125?

Write the reciprocal of 4 as a decimal.

You must show all your approximations in your working.

Evaluate

Give your answer in its simplest form.

What is the greatest integer value of n if 2n < 17?

What is the least integer value of n if 2ⁿ > 125?

The diagram below shows two shaded squares inside a larger square.

The diagram shows the area of each of the two shaded squares.

Calculate the total area of the two regions that have not been shaded.

You must show all your working.

A decimal number is written on a card.

You have three clues to help you work out the number on the card. Clue 1: The number is between 5 and 12 inclusive. Clue 2: The number is a multiple of 2·3. Clue 3: The square of the number is greater than 50 but less than 120.

What is the decimal number on the card?

The Geometric Mean is a special type of average.

To find the Geometric Mean of two numbers, you must:

• multiply the two numbers together, and

• then find the square root.

Find the Geometric Mean of 250 and 0·4.

To find the Geometric Mean of three numbers, you must:

• multiply the three numbers together, and

• then find the cube root.

Find the Geometric Mean of 100, 0·3 and 0·9.

The Geometric Mean of three numbers is 10. Two of the numbers are 8 and 25.

Find the third number.

A decimal number is written on a card.

You have three clues to help you work out the number on the card. Clue 1: The number is between 5 and 12 inclusive. Clue 2: The number is a multiple of 2·3. Clue 3: The square of the number is greater than 50 but less than 120.

What is the decimal number on the card?

The Geometric Mean is a special type of average.

To find the Geometric Mean of three numbers, you must:

• multiply the three numbers together, and

• then find the cube root.

Find the Geometric Mean of 100, 0·3 and 0·9.

The Geometric Mean of three numbers is 10. Two of the numbers are 8 and 25.

Find the third number.

Circle the correct answer for each of the following statements.

Give an example of an irrational number that lies between 6 and 7.

My example of an irrational number is_........

Billy thinks of a number. Billy halves his number. His answer is 58.

What number did Billy think of?

Siân thinks of a different number. Siân squares her number. Her answer is 9.

What number did Siân think of?

Calculate 40% of 120.

Express 48 as a percentage of 400.

Share £45 in the ratio 8: 1.

£_........ and £_........

Express 1 as a single fraction in the form , where a and b are integers.

Give your answer correct to 2 significant figures.

Calculate the reciprocal of 23.

Give your answer correct to 3 decimal places.

Circle the correct answer for each of the following.

The Lowest Common Multiple (LCM) of 4 and 6 is:

2 4 6 12 24

The Highest Common Factor (HCF) of 10 and 15 is:

5 10 15 30 150

Write down an integer value of x that is greater than 5, for which and

are both rational.

Give your answer correct to 2 significant figures.

Calculate the reciprocal of 23.

Give your answer correct to 3 decimal places.

Circle the correct answer for each of the following.

The Lowest Common Multiple (LCM) of 4 and 6 is:

2 4 6 12 24

The Highest Common Factor (HCF) of 10 and 15 is:

5 10 15 30 150

The cube shown below has a volume of 10 648 cm³.

Calculate the length of the internal diagonal AB.

Two friends, Geraint and Dyfrig, are having a discussion.

Explain why Geraint is incorrect.

Dyfrig says,

“All cube numbers are odd numbers.”

Explain why Dyfrig is incorrect.

Explain why Geraint is incorrect.

Dyfrig says,

“All cube numbers are odd numbers.”

Explain why Dyfrig is incorrect.

A rectangle and a square are shown below.

The total area of the two shapes is 184 cm².

Find the perimeter of the square.

Simplify 12a 19a 10a.

Solve the following equations.

Calculate each of the following.

3² 2³

124 4

15% of 280

Elaine writes down two square numbers.

She subtracts the smaller square number from the larger square number. Her answer is 9.

Which two square numbers did Elaine write down?

Elaine’s square numbers are_........ and_........

Calculate each of the following.

3² 2³

124 4

15% of 280

Simplify your answer.

Which one of these numbers is both a square number and an even number?

Circle the correct answer.

2 9 12 16 17

Write 75% as a fraction in its lowest terms.

Write down the mode of these numbers.

28 31 28 29 31 28 34 24 32

Give your answer in its simplest form.

Calculate the value of 3³ ÷ 2².

Give your answer as a decimal.

A number n is added to the square root of 81. The answer is equal to 7 squared.

What is the value of n?

Give your answer in its simplest form.

Calculate the value of 3³ ÷ 2².

Give your answer as a decimal.

A number n is added to the square root of 81. The answer is equal to 7 squared.

What is the value of n?

Calculate each of the following.

3³ × 10²

0·4 × 0·2

Evaluate 5·4³ × 3·7².

Give your answer correct to the nearest 10.

Find 62% of 7·8.

Which one of the following numbers is a multiple of 19?

Circle your answer.

91 151 199 219 247

Which one of the following numbers is a cube number?

Circle your answer.

1197 2197 3197 4197 5197

Calculate each of the following.

3³ × 10²

0·4 × 0·2

Write down the value of 0·0493, correct to 1 significant figure.

Evaluate 5·4³ × 3·7².

Give your answer correct to the nearest 10.

Find 62% of 7·8.

Which one of the following numbers is a multiple of 19?

Circle your answer.

91 151 199 219 247

Which one of the following numbers is a cube number?

Circle your answer.

1197 2197 3197 4197 5197

Match each expression with its value.

The first one has been done for you.

Space for working:

Find the whole number which satisfies all of the following conditions:

• It is a whole number between 1 and 40 inclusive.

• The number is a multiple of 4 but not a multiple of 8.

• 3 is a factor of this number.

• The number is a square number.

The whole number is_{……………........ ………..}

Find the whole number which satisfies all of the following conditions:

• It is a whole number between 1 and 40 inclusive.

• The number is a multiple of 4 but not a multiple of 8.

• 3 is a factor of this number.

• The number is a square number.

The whole number is_{……………........ ………..}

Circle the correct answer for each of the following.

81 =

2·1⁵ =

32·5 10·5 40·84101 30·84101 32·1

6·48 3·6 4·32 3·3 2·16

Circle the correct answer for each of the following.

81 =

2·1⁵ =

32·5 10·5 40·84101 30·84101 32·1

6·48 3·6 4·32 3·3 2·16

Write 481·627 correct to 2 decimal places.

Write down the value of 8². (b)

Work out 38·25 ÷ 1000.

Bethan writes down two square numbers.

She adds her two numbers together. Her answer is a square number less than 30.

Which two square numbers did Bethan write down?

Bethan’s square numbers are_{……........ ……} and_{……........ ……}

Harri adds three even numbers together and gets an answer of 23.

Explain how you know that Harri’s answer is incorrect.

Calculate each of the following.

1000 × (the reciprocal of 8)

Write 437·6 correct to 2 significant figures.

Place the numbers 1, 2, 3, 4, 5 in the Venn diagram below.

Space for working:

Consider the following list of numbers:

4 25 27 36 49 64 90 125

Using only the numbers in the list above, write down

the three cube numbers,

a number that is both a square number and a multiple of 9,

a number that is a factor of 81.

Using only two numbers from the list, fill in the spaces in the following statement:

‘Dividing_........ by_......... and then rounding the answer to the nearest 10, gives an answer of 30.’

Share £720 in the ratio 2: 7.

Calculate the value of the reciprocal of 0.2. (b)

Circle the correct answer for each of the following statements.

35² 15² 15 35 15⁴

3^{– 4} is equal to

Jonathan writes a whole number on a card.

• The number is between 55 and 83 inclusive.

• The square root of the number is greater than 8 but less than 9.

• The highest common factor (HCF) of 49 and the number is 7.

What number is on Jonathan’s card?

The number on Jonathan’s card is_{………........ ……………..}

Calculate the value of the reciprocal of 0.2. (b)

Circle the correct answer for each of the following statements.

35² 15² 15 35 15⁴

3^{– 4} is equal to

Jonathan writes a whole number on a card.

• The number is between 55 and 83 inclusive.

• The square root of the number is greater than 8 but less than 9.

• The highest common factor (HCF) of 49 and the number is 7.

What number is on Jonathan’s card?

The number on Jonathan’s card is_{………........ ……………..}

Find the whole number that satisfies all of the following conditions.

• It is a whole number between 1 and 100 inclusive. • 10% of the number is greater than 2 but less than 8. • of the number is a square number.

• The number is not a multiple of 4.

The Venn diagram below is used for showing square numbers and even numbers.

Place the numbers 1, 2, 3, 4 and 5 in the Venn diagram.

Find the whole number that satisfies all of the following conditions.

• It is a whole number between 1 and 100 inclusive. • 10% of the number is greater than 2 but less than 8. • of the number is a square number.

• The number is not a multiple of 4.

Calculate the reciprocal of 47, giving your answer correct to 4 decimal places.

The highest common factor (HCF) of 30 and 75 is the square root of a number.

What is the number?

The cube root of 32.768 is 33 % of a number.

What is the number?

Calculate the reciprocal of 47, giving your answer correct to 4 decimal places.

The highest common factor (HCF) of 30 and 75 is the square root of a number.

What is the number?

The cube root of 32.768 is 33 % of a number.

What is the number?

Calculate each of the following.

3⁴ × 10³

5.6 − 3.82

0.2 × 0.3

Write down the value of 9². 8. (a)

Work out 1.2 70.

Express 700 as a product of its prime factors in index form.

(b)

Explain how this tells you that 33 554 432 is not a square number.

Calculate the reciprocal of −0.07, correct to 1 decimal place.

Express 700 as a product of its prime factors in index form.

(b)

Explain how this tells you that 33 554 432 is not a square number.

Calculate the reciprocal of −0.07, correct to 1 decimal place.