GCSEMathsOCRExam Questions3. Indices & SurdsPowers, Roots & Standard Form

Powers, Roots & Standard Form (OCR GCSE Maths)

Exam Questions

3 hours53 questions
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Hard
1
Sme Calculator2 marks

Write these numbers in order of size.
Start with the smallest number.

5 to the power of negative 1 end exponent 0.5 negative 5 5 to the power of 0

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2
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Write down the value of  125 to the power of 2 over 3 end exponent.

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3
Sme Calculator1 mark

Evaluate 3 to the power of negative 2 end exponent.

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4a
Sme Calculator1 mark

Writespace 7.8 cross times 10 to the power of negative 4 end exponent as an ordinary number.

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4b
Sme Calculator1 mark

Write 95 space 600 space 000 as a number in standard form.

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5
Sme Calculator2 marks

Work out the value of space left parenthesis 7.5 space cross times 10 to the power of 4 right parenthesis space cross times space left parenthesis 2.5 space cross times space 10 cubed right parenthesis space
Give your answer in standard form.

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6
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Write 0.000068 in standard form.

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7a
Sme Calculator1 mark

Write 0.000423 in standard form.

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7b
Sme Calculator1 mark

Write 4.5 space cross times space 10 to the power of 4 as an ordinary number.

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8a
Sme Calculator1 mark

Write space 0.00562 spacein standard form.

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8b
Sme Calculator1 mark

Write 1.452 space cross times space 10 cubed as an ordinary number.

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9
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Write down the reciprocal of 5.

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10a
Sme Calculator2 marks

Write these numbers in standard form.

i)
6500

[1]
ii)
0.0584
[1]

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10b
Sme Calculator1 mark

Work out open parentheses 4.2 space cross times 10 to the power of 5 close parentheses cross times open parentheses 1.8 cross times 10 to the power of negative 2 end exponent close parentheses giving your answer in standard form.

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11
Sme Calculator2 marks

Work out open parentheses 2 cross times 10 cubed close parentheses cross times space open parentheses 4 cross times 10 to the power of 4 close parentheses, giving your answer in standard form.

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12
Sme Calculator1 mark

Evaluate.

8 to the power of begin inline style 1 third end style end exponent

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13a
Sme Calculator2 marks

A grain of salt weighs 6.48 space cross times space 10 to the power of negative 5 end exponent spacekg on average.

A packet contains 0.35 kg of salt.


Use this information to calculate the number of grains of salt in the packet.

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13b
Sme Calculator1 mark

Explain why your answer to part (a) is unlikely to be the actual number of grains of salt in the packet.

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14
Sme Calculator4 marks

Tom researches the weights of plant seeds.

  • One poppy seed weighs 3 cross times 10 to the power of negative 4 end exponent grams
  • 250 pumpkin seeds weigh 21 grams.
  • One sesame seed weighs 3.64 cross times 10 to the power of negative 6 end exponent kilograms.

Write the three types of seed in order according to the weight of one seed.
Write the lightest type of seed first.
You must show how you decide.

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15
Sme Calculator5 marks

A newborn baby has an approximate mass of 3.5 kilograms.

A human cell has an approximate mass of 2.7 × 10–11 grams.

Use these values to estimate the number of human cells in a newborn baby.
Give your answer in standard form, correct to 2 significant figures.

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16a
Sme Calculator1 mark

Write down the value of m, given that  3 to the power of 4 space end exponent cross times space space 3 to the power of 5 space equals space 3 to the power of m

m space equals.................................. 

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16b
Sme Calculator1 mark

Write down the value of n, given that open parentheses 5 cubed close parentheses to the power of 7 equals 5 to the power of n

n space equals.............................

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16c
Sme Calculator2 marks

Find the value of p, given that fraction numerator 7 to the power of 8 cross times 7 squared over denominator 7 to the power of p end fraction equals 7 to the power of 6

p space equals .......................................................

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17
Sme Calculator1 mark

Write 5 to the power of 17 space cross times space 5 squared as a single power of 5.

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18a
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Write 2.46 cross times 10 to the power of 6 as an ordinary number.

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18b
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Write 0.000 space 74 in standard form.

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19
Sme Calculator1 mark

Which one of these is a square number and a cube number?

Circle your answer.

100 1000 10 000 1 000 000

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20
Sme Calculator1 mark

Circle the number that is written in standard form.

0.9 × 10–3 6 × 100.5 5.2 × 10–4 12 × 107

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1
Sme Calculator3 marks

One sheet of paper is  9 space cross times space 10 to the power of negative 3 end exponentcm thick.

Mark wants to put space 500 spacesheets of paper into the paper tray of his printer.
The paper tray is 4 cm deep.

Is the paper tray deep enough for 500 sheets of paper?
You must explain your answer.

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2
Sme Calculator2 marks

Write the following numbers in order of size.
Start with the smallest number.

0.038 space cross times space 10 squared 3800 space cross times space 10 to the power of negative 4 end exponent 380 0.38 space cross times space 10 to the power of negative 1 end exponent

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3a
Sme Calculator1 mark

Write down the value of 10 to the power of 0.

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3b
Sme Calculator1 mark

Write down the value of 10 to the power of negative 2 end exponent.

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3c
Sme Calculator2 marks

Write these numbers in order of size.
Start with the smallest number.

2.73 space cross times space 10 cubed 27.3 space cross times space 10 to the power of negative 3 end exponent 273 space cross times space 10 squared 0.00273

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4a
Sme Calculator1 mark

Write down the value of  100 to the power of 1 half end exponent.

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4b
Sme Calculator2 marks

Find the value of 125 to the power of 2 over 3 end exponent.

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5a
Sme Calculator1 mark

Write down the value of 36 to the power of 1 half end exponent.

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5b
Sme Calculator1 mark

Write down the value of 23 to the power of 0.

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5c
Sme Calculator2 marks

Work out the value of 27 to the power of negative 2 over 3 end exponent.

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6a
Sme Calculator1 mark

The table shows some information about eight planets.

Planet Distance from Earth (km) Mass (kg)
Earth 0 5.97 cross times 10 to the power of 24
Jupiter 6.29 cross times 10 to the power of 8 1.898 cross times 10 to the power of 27
Mars 7.83 cross times 10 to the power of 7 6.42 cross times 10 to the power of 23
Mercury 9.17 cross times 10 to the power of 7 3.302 cross times 10 to the power of 23
Neptune 4.35 space cross times space 10 to the power of 9 1.024 space cross times space 10 to the power of 26
Saturn 1.28 space cross times space 10 to the power of 9 5.68 space cross times space 10 to the power of 26
Uranus 2.72 space cross times space 10 to the power of 9 8.683 space cross times space 10 to the power of 25
Venus 4.14 space cross times space 10 to the power of 7 4.869 space cross times space 10 to the power of 24

Write down the name of the planet with the greatest mass.

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6b
Sme Calculator1 mark

Find the difference between the mass of Venus and the mass of Mercury.

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6c
Sme Calculator2 marks

Nishat says that Neptune is over a hundred times further away from Earth than Venus is.

Is Nishat right?
You must show how you get your answer.

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7
Sme Calculator2 marks

Work out left parenthesis 13.8 space cross times space 10 to the power of 7 right parenthesis space cross times space left parenthesis 5.4 cross times 10 to the power of negative 12 end exponent right parenthesis
Give your answer as an ordinary number.

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8a
Sme Calculator1 mark

Write down the value of 100.

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8b
Sme Calculator1 mark

Write 6.7 x 10-5 as an ordinary number.

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8c
Sme Calculator2 marks

Work out the value of (3 x 107) x (9 x 106).

Give your answer in standard form.

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9
Sme Calculator2 marks

Calculate 9 space cross times 10 to the power of 4 space cross times space 3 space cross times space 10 cubed.
Give your answer in standard form.

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10
Sme Calculator2 marks

Work out the value of left parenthesis 9 space cross times 10 to the power of negative 4 end exponent right parenthesis space cross times space left parenthesis 3 space cross times 10 to the power of 7 right parenthesis.
Give your answer in standard form.

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11a
Sme Calculator1 mark

Work out the value of 25-3.

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11b
Sme Calculator2 marks

Work out the value of 3503.
Give your answer in standard form.

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12
Sme Calculator1 mark

Patrick has to work out the exact value of 64 to the power of 1 fourth end exponent

Patrick says,

      "1 fourth of 64 is 16 so 64 to the power of 1 fourth end exponent space equals space 16"

Explain what is wrong with what Patrick says.

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13
Sme Calculator3 marks

Write open parentheses fourth root of 8 close parentheses to the power of 5 as a power of 2.

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14
Sme Calculator3 marks

You are given that 177147 equals 3 to the power of 11

3 to the power of n equals 177147 space cross times 9 to the power of 5

Find the value of n.

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15a
Sme Calculator2 marks

Work out.

cube root of 64 space cross times space 2 to the power of negative 1 end exponent

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15b
Sme Calculator3 marks

4.3 cross times 10 to the power of 5 space plus 3.8 space cross times 10 to the power of 4

Give your answer in standard form.

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16
Sme Calculator3 marks

Carol says that 64 to the power of negative 1 half end exponent space equals space 1 over 32

Explain her error and give the correct value of 64 to the power of negative 1 half end exponent spacein the form p over q.

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17a
Sme Calculator2 marks

A company makes sweets.
The sweets are put into packets.

Here are some facts.

bold 1 bold. bold 47 bold space bold cross times bold space bold 10 to the power of bold 7

sweets are made every day

  

bold 3 bold. bold 5 bold space bold cross times bold space bold 10 to the power of bold 5

packets of sweets are 
produced every day

Calculate the mean number of sweets in one packet.

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17b
Sme Calculator3 marks

Sweets are made on 288 days each year.

Calculate the number of sweets made each year.
Give your answer in standard form.

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17c
Sme Calculator4 marks

The company has 152 machines making the sweets.
Each machine operates for 15 hours each day.

i)
Calculate the number of sweets made by one machine each hour.
Give your answer as an ordinary number correct to the nearest 10.

[3]

ii)
State one assumption you have made in part (c)(i).

[1]

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18a
Sme Calculator2 marks

The table shows information about the surface area of each of the world’s oceans.

Ocean Surface area in
square kilometres
  Pacific 1.56 space cross times space 10 to the power of 8
  Indian 6.86 space cross times space 10 to the power of 7
  Southern 2.03 space cross times space 10 to the power of 7
  Arctic 1.41 space cross times space 10 to the power of 7
  Atlantic 1.06 space cross times space 10 to the power of 8

Work out the difference, in square kilometres, between the surface area of the Atlantic Ocean and the surface area of the Indian Ocean.
Give your answer in standard form.

.......................square kilometres

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18b
Sme Calculator1 mark

The surface area of the Pacific Ocean is k times the surface area of the Arctic Ocean.

Work out the value of k.
Give your answer correct to the nearest whole number.

k space equals...........................

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1a
Sme Calculator1 mark

Find the value of 2 to the power of negative 3 end exponent.

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1b
Sme Calculator1 mark

5 square root of 5 can be written in the form 5 to the power of k.

Find the value of k.

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2a
Sme Calculator2 marks

Find the value of 81 to the power of negative 1 half end exponent.

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2b
Sme Calculator2 marks

Find the value of open parentheses 64 over 125 close parentheses to the power of 2 over 3 end exponent.

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3a
Sme Calculator1 mark

Write down the value of 64 to the power of 1 half end exponent.

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3b
Sme Calculator2 marks

Find the value of open parentheses 8 over 125 close parentheses to the power of negative 2 over 3 end exponent.

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4a
Sme Calculator1 mark

Find the value of cube root of 8 cross times 10 to the power of 6 end root.

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4b
Sme Calculator2 marks

Find the value of     144 to the power of 1 half end exponent cross times 64 to the power of negative 1 third end exponent.

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4c
Sme Calculator2 marks

Solve     3 to the power of 2 x end exponent space equals fraction numerator space 1 over denominator 81 end fraction.

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5a
Sme Calculator1 mark

Write 640 space 000 space 000 in standard form.

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5b
Sme Calculator2 marks

Work out left parenthesis 3 space cross times space 10 to the power of 7 right parenthesis space divided by space left parenthesis 6 space cross times space 10 to the power of 4 right parenthesis
Give your answer in standard form.

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6a
Sme Calculator1 mark

Write 5 space 400 space 000 as a number in standard form.

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6b
Sme Calculator1 mark

Write space 3.2 space cross times space 10 to the power of negative 4 end exponent spaceas an ordinary number.

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6c
Sme Calculator2 marks

The mass of the Sun is 2 cross times space 10 to the power of 30 kg.
The mass of the largest known star is space 315 spacetimes the mass of the Sun.

Work out the mass of this star.
Give your answer in kg in standard form.

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7a
Sme Calculator1 mark

Write 7.97 space cross times space 10 to the power of negative 6 end exponent as an ordinary number.

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7b
Sme Calculator2 marks

Work out the value of left parenthesis 2.52 space cross times space 10 to the power of 5 right parenthesis space divided by space left parenthesis 4 space cross times 10 to the power of negative 3 end exponent right parenthesis
Give your answer in standard form.

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8
Sme Calculator3 marks

      p squared equals space fraction numerator x minus y over denominator x y end fraction

x equals 8.5 cross times 10 to the power of 9 y equals 4 cross times 10 to the power of 8

Find the value of p.
Give your answer in standard form correct to 2 significant figures.

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9a
Sme Calculator2 marks

T space equals space square root of w over d cubed end root

w space equals space 5.6 cross times 10 to the power of negative 5 end exponent d space space equals space 1.4 cross times 10 to the power of negative 4 end exponent space

Work out the value of T.
Give your answer in standard form correct to 3 significant figures.

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9b
Sme Calculator2 marks

w is increased by 10 percent sign
d is increased by 5 percent sign

Lottie says,
         "The value of T will increase because both w and d are increased."

Lottie is wrong.
Explain why.

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10a
Sme Calculator1 mark

Write 8.2 x 105 as an ordinary number.

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10b
Sme Calculator1 mark

Write 0.000 376 in standard form.

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10c
Sme Calculator2 marks

Work out the value of (2.3 x 1012) ÷ (4.6 x 103)
Give your answer in standard form.

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11
Sme Calculator2 marks

Work out the value of (3.5 x  106) ÷ (5 x 10-3).
Give your answer in standard form.

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12
Sme Calculator3 marks

There are two errors in Sam’s method for finding the value of 64 to the power of begin inline style fraction numerator 2 presuperscript minus over denominator 3 end fraction end style end exponentshown 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 to the power of fraction numerator 2 presuperscript minus over denominator 3 end fraction end exponent.

Correct value ........................................................

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13
Sme Calculator3 marks

Show that  fraction numerator cube root of 81 over denominator 3 end fraction can be written as 3 to the power of 1 third end exponent

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14
Sme Calculator3 marks

Simplify  8 squared space cross times cube root of 4 to the power of 6 end root

Give your answer in the form 2a where a is an integer.

Show each stage of your working clearly.

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15
Sme Calculator3 marks

a space equals space 25 space cross times space 10 to the power of 14 n end exponent where nis an integer.

Find an expression, in terms of n , for a to the power of 3 over 2 end exponent

Give your answer in standard form.

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