Teach Me Maths

Standard Form

Question 11 mark

Write 6.28\times10^4 as an ordinary number.

Hint

Multiplying by 10 four times moves every digit four places to the left.

Worked solution

Write 6.28\times10^4 as an ordinary number.

  1. 10^4=10000.
  2. 6.28\times10000=62800.

Question 21 mark

Write 3070000 in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Choose a coefficient at least 1 and less than 10, then count the place-value change.

Worked solution

Write 3070000 in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

  1. The coefficient is 3.07. Multiplying it by 10^6 gives 3070000.
  2. Answer: 3.07\times10^6.

Question 31 mark

Write 4.36\times10^{-3} as an ordinary number.

Hint

A negative power of 10 makes a positive number smaller.

Worked solution

Write 4.36\times10^{-3} as an ordinary number.

  1. 10^{-3}=\frac{1}{1000}.
  2. 4.36\div1000=0.00436.

Question 41 mark

Write 0.0000712 in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Start with 7.12. Decide how many divisions by 10 give the original small number.

Worked solution

Write 0.0000712 in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

  1. 7.12\div100000=0.0000712.
  2. Dividing by 10^5 is multiplying by 10^{-5}. Answer: 7.12\times10^{-5}.

Question 51 mark

Write 8.4\times10^0 as an ordinary number.

Hint

Any non-zero number to the power 0 is 1.

Worked solution

Write 8.4\times10^0 as an ordinary number.

  1. 10^0=1, so 8.4\times10^0=8.4\times1=8.4.

Question 61 mark

Which expression is in standard form?

Hint

Check the coefficient and check that the power of 10 is an integer (positive, negative or zero).

Worked solution

Which expression is in standard form?

  1. 73\times10^2
  2. 7.3\times10^3
  3. 0.73\times10^4
  4. 7.3\times10^{1.5}
  1. 7.3\times10^3 has a coefficient at least 1 and less than 10 and an integer exponent.
  2. The coefficient 73 is too large, 0.73 is too small, and 1.5 is not an integer exponent.

Question 71 mark

Write 4005000 in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Keep the zeros between the significant digits.

Worked solution

Write 4005000 in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

  1. The coefficient must be 4.005, preserving both zeros.
  2. 4.005\times1000000=4005000, so the answer is 4.005\times10^6.

Question 81 mark

Which is the greatest number?

Hint

Convert the values to ordinary decimals, or compare their powers carefully.

Worked solution

Which is the greatest number?

  1. 2.7\times10^{-3}
  2. 8.1\times10^{-4}
  3. 1.9\times10^{-2}
  4. 6.3\times10^{-3}
  1. The ordinary values are 0.0027, 0.00081, 0.019 and 0.0063.
  2. The greatest is 0.019, which is 1.9\times10^{-2}.

Question 92 marks

Work out 6.4\times10^5+5.1\times10^5. Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

The powers match, so add the coefficients first.

Worked solution

Work out 6.4\times10^5+5.1\times10^5. Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

  1. (6.4+5.1)\times10^5=11.5\times10^5.
  2. The coefficient 11.5 is too large. 11.5\times10^5=1.15\times10^6.

Question 102 marks

Work out 3\times(7.4\times10^{-4}). Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Multiply the coefficient, then check that it is at least 1 and less than 10.

Worked solution

Work out 3\times(7.4\times10^{-4}). Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

  1. 3\times7.4=22.2, so the product is 22.2\times10^{-4}.
  2. 22.2\times10^{-4}=2.22\times10^{-3}.

Question 112 marks

Work out 8.6\times10^5+4.7\times10^4. Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

The powers differ. Ordinary numbers are a useful way to add them.

Worked solution

Work out 8.6\times10^5+4.7\times10^4. Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

  1. 8.6\times10^5=860000 and 4.7\times10^4=47000.
  2. 860000+47000=907000=9.07\times10^5.

Question 122 marks

Work out 6.2\times10^{-3}-8.5\times10^{-4}. Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Write the two values as ordinary decimals before subtracting.

Worked solution

Work out 6.2\times10^{-3}-8.5\times10^{-4}. Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

  1. 0.0062-0.00085=0.00535.
  2. 0.00535=5.35\times10^{-3}.

Question 132 marks

Work out (3\times10^4)\times(8\times10^{-7}). Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Multiply the coefficients and combine the powers, then check the form.

Worked solution

Work out (3\times10^4)\times(8\times10^{-7}). Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

  1. 3\times8=24 and 10^4\times10^{-7}=10^{-3}.
  2. 24\times10^{-3}=0.024=2.4\times10^{-2}.

Question 142 marks

Work out \dfrac{8.4\times10^7}{2.1\times10^3}. Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Divide the coefficients and the powers of 10 separately.

Worked solution

Work out \dfrac{8.4\times10^7}{2.1\times10^3}. Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

  1. 8.4\div2.1=4 and 10^7\div10^3=10^4.
  2. Answer: 4\times10^4.

Question 152 marks

An archive contains 4.8\times10^5 images. Each image uses 2.5 megabytes. Work out the total number of megabytes. Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Find the total storage used by all the images.

Worked solution

An archive contains 4.8\times10^5 images. Each image uses 2.5 megabytes. Work out the total number of megabytes. Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

  1. Total storage: (4.8\times10^5)\times2.5=12\times10^5 megabytes.
  2. 12\times10^5=1.2\times10^6 megabytes.

Question 16Challenge3 marks

A piece of wire is 6.4\times10^{-2} metres long. It is cut into pieces, each 8\times10^{-4} metres long. There is no waste. Five of the pieces are rejected. How many pieces remain?

Hint

Find the number cut before taking away the rejected pieces.

Worked solution

A piece of wire is 6.4\times10^{-2} metres long. It is cut into pieces, each 8\times10^{-4} metres long. There is no waste. Five of the pieces are rejected. How many pieces remain?

  1. Length of wire: 0.064 m. Length of each piece: 0.0008 m.
  2. Number cut: 0.064\div0.0008=80.
  3. Number remaining: 80-5=75.

Question 17Challenge3 marks

An empty container has a mass of 1.8\times10^3 grams. The container and its contents have a total mass of 7.56\times10^3 grams. The contents are identical parts, each of mass 2.4\times10^1 grams. How many parts are in the container?

Hint

Remove the mass of the container before finding how many parts it holds.

Worked solution

An empty container has a mass of 1.8\times10^3 grams. The container and its contents have a total mass of 7.56\times10^3 grams. The contents are identical parts, each of mass 2.4\times10^1 grams. How many parts are in the container?

  1. Mass of the contents: 7560-1800=5760 g.
  2. Each part has mass 2.4\times10^1=24 g.
  3. Number of parts: 5760\div24=240.

Question 18Challenge4 marks

Tank A holds 1.44\times10^6 litres. Tank B holds 3.6\times10^4 litres.

(a)

How many times as much water can tank A hold as tank B?

2 marks

(b)

Tank A is empty and tank B is full. All the water from B is pumped into A. How many more litres are needed to fill A? Give your answer in standard form. Use x or an asterisk (*) for multiplication and ^ for the power.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Use the same units in both parts. Decide whether each part needs a division or a subtraction.

Worked solution

Tank A holds 1.44\times10^6 litres. Tank B holds 3.6\times10^4 litres.

Part (a)

How many times as much water can tank A hold as tank B?

  1. 1440000\div36000=40. Tank A holds 40 times as much.

Part (b)

Tank A is empty and tank B is full. All the water from B is pumped into A. How many more litres are needed to fill A? Give your answer in standard form. Use x or an asterisk (*) for multiplication and ^ for the power.

  1. Water still needed: 1440000-36000=1404000 litres.
  2. Answer: 1.404\times10^6 litres.

Question 19Challenge3 marks

Nina works out (6\times10^{-3})\times(4\times10^{-5}). She gets 24\times10^{-8} and then writes 2.4\times10^{-8}.

(a)

Give the correct final answer in standard form. Use x or an asterisk (*) for multiplication and ^ for the power.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Which statement explains her mistake?

1 mark

Hint

Changing the coefficient also requires a change to the power if the value is to stay the same.

Worked solution

Nina works out (6\times10^{-3})\times(4\times10^{-5}). She gets 24\times10^{-8} and then writes 2.4\times10^{-8}.

Part (a)

Give the correct final answer in standard form. Use x or an asterisk (*) for multiplication and ^ for the power.

  1. 6\times4=24 and 10^{-3}\times10^{-5}=10^{-8}.
  2. 24\times10^{-8}=2.4\times10^{-7}.

Part (b)

Which statement explains her mistake?

  1. She should subtract the two exponents when multiplying.
  2. The first product should have coefficient 10.
  3. A negative exponent always makes the whole answer negative.
  4. She changed the coefficient without increasing the exponent by 1.
  1. Dividing the coefficient by 10 requires multiplying the power-of-ten factor by 10.
  2. So the exponent must increase from -8 to -7.

Question 20Challenge3 marks

A culture starts with 3.6\times10^5 cells. The number of cells doubles each hour for 3 hours. Then 4\times10^5 cells are removed. How many cells remain? Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Three doublings multiply the starting number by 2 three times. The removal happens afterwards.

Worked solution

A culture starts with 3.6\times10^5 cells. The number of cells doubles each hour for 3 hours. Then 4\times10^5 cells are removed. How many cells remain? Give your answer in standard form.

Use x or an asterisk (*) for multiplication and ^ for the power.

  1. After 3 hours: (3.6\times10^5)\times2^3=360000\times8=2880000 cells.
  2. After removal: 2880000-400000=2480000 cells.
  3. Answer: 2.48\times10^6 cells.

More on this topic: Standard Form worksheet with full solutions

All Edexcel GCSE Foundation practice questions