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Standard Form

Standard Form

Calculating in standard form

Dmas Paper 1 + 2 12 min read

On this page
  1. In this note
  2. Formula bank
  3. Tidying up: getting A between 1 and 10
  4. Multiplying
  5. Dividing
  6. Adding
  7. Subtracting
  8. Squares, cubes, roots and reciprocals
  9. On the calculator paper
  10. Ways they test it
  11. Mistakes to avoid
  12. Summary
  13. Practice
  14. Can I do this?
  15. Quiz

In this note

Syllabus 1.8: calculating with standard form
  • 1.8Multiply and divide numbers in standard form without a calculator.
  • 1.8Add and subtract them (the powers of 10 must match first).
  • 1.8Square, cube, square-root and find the reciprocal of a number in standard form.
  • 1.8Use the calculator correctly and give the answer back in standard form.

Before you start: writing numbers in standard form and the index laws. 0580 Core: calculating with standard form is on the calculator paper only.

Formula bank

Standard formLearn it\(A\times10^{n}\), \(1\le A<10\)\(A\) = a number from 1 up to (not including) 10; \(n\) = a whole number.Use it when the question says "give your answer in standard form".
MultiplyLearn it\((A\times10^{m})(B\times10^{n})=AB\times10^{m+n}\)multiply the numbers, ADD the powers.Use it when two standard form numbers are multiplied.
DivideLearn it\(\dfrac{A\times10^{m}}{B\times10^{n}}=\dfrac AB\times10^{m-n}\)divide the numbers, SUBTRACT the powers.Use it when one is divided by the other.
Add / subtractLearn it\(A\times10^{n}\pm B\times10^{n}=(A\pm B)\times10^{n}\)only once the powers are the SAME.Use it when a sum or difference: match the powers first.
Power of a numberLearn it\((A\times10^{n})^{k}=A^{k}\times10^{nk}\)the power goes on BOTH parts.Use it when squaring or cubing.
Square rootLearn it\(\sqrt{A\times10^{n}}=\sqrt A\times10^{n/2}\)make \(n\) even first.Use it when a square root without a calculator.
Tidy upLearn it\(A\div10\Rightarrow n+1\qquad A\times10\Rightarrow n-1\)the number and the power move in opposite directions.Use it when your \(A\) is 10 or more, or less than 1.

Tidying up: getting A between 1 and 10

Every calculation below ends the same way: the number part must be from 1 to just under 10. If it isn't, move the decimal point and change the power to make up for it. Make the number 10 times smaller, and the power goes up by 1, so the value stays the same.

General methodHow to tidy any answer into standard form

  1. Look at the number part \(A\). Is it from 1 to just under 10? Then you are done.
  2. Too big (10 or more)? Move the decimal point LEFT. For each place moved, ADD 1 to the power.
  3. Too small (less than 1)? Move the decimal point RIGHT. For each place moved, SUBTRACT 1 from the power.

Example Write \(16.8\times10^{18}\) in standard form.

16.8 is too big: move the point one place left, so the power goes up by 1.

  1. \(16.8=1.68\times10^{1}\)
  2. \(1.68\times10^{1}\times10^{18}=1.68\times10^{19}\)

Answer \(1.68\times10^{19}\)

Try it yourself

1
Write \(35\times10^{5}\) in standard form.
Show answer

Answer \(3.5\times10^{6}\)

2
Write \(0.4\times10^{-5}\) in standard form.
Hint
0.4 is too small: point one place right, power down by 1.
Show answer

Answer \(4\times10^{-6}\)

Multiplying

When you multiply, the order does not matter, so put the numbers together and the powers of 10 together: \((4\times10^{5})\times(2\times10^{-3})=(4\times2)\times(10^{5}\times10^{-3})\).

General methodHow to multiply any two numbers in standard form

  1. Multiply the number parts: \(A\times B\).
  2. ADD the powers of 10: \(10^{m}\times10^{n}=10^{m+n}\) (watch the signs with negative powers).
  3. Tidy up so the number part is from 1 to just under 10.

Example Work out \((2.1\times10^{8})\times(8\times10^{10})\).

Multiply the numbers, add the powers, then tidy.

  1. \(2.1\times8=16.8\)
  2. \(10^{8+10}=10^{18}\)
  3. \(16.8\times10^{18}=1.68\times10^{19}\)

Answer \(1.68\times10^{19}\)

Try it yourself

1
\((3\times10^{4})\times(2\times10^{5})\)
Show answer

Answer \(6\times10^{9}\)

2
\((5\times10^{6})\times(4\times10^{3})\)
Hint
\(5\times4=20\): tidy it.
Show answer

Answer \(2\times10^{10}\)

Dividing

General methodHow to divide any two numbers in standard form

  1. Divide the number parts: \(A\div B\) (top number ÷ bottom number).
  2. SUBTRACT the powers: top power − bottom power. Subtracting a negative power ADDS: \(9-(-3)=12\).
  3. Tidy up. A small answer like \(0.4\) becomes \(4\) and the power goes down by 1.

Example Work out \(\dfrac{2\times10^{3}}{5\times10^{8}}\).

Divide the numbers, subtract the powers, then tidy.

  1. \(2\div5=0.4\)
  2. \(10^{3-8}=10^{-5}\)
  3. \(0.4\times10^{-5}=4\times10^{-6}\)

Answer \(4\times10^{-6}\)

Try it yourself

1
\(\dfrac{9\times10^{8}}{3\times10^{2}}\)
Show answer

Answer \(3\times10^{6}\)

2
\(\dfrac{3\times10^{5}}{6\times10^{-2}}\)
Hint
\(3\div6=0.5\), and \(5-(-2)=7\). Then tidy.
Show answer

Answer \(5\times10^{6}\)

Adding

You can NOT add the number parts and the powers separately: \(3\times10^{4}+2\times10^{3}\) is \(30\,000+2000=32\,000\), not \(5\times10^{7}\). The powers have to match first, like adding fractions needs a common denominator.

General methodHow to add any two numbers in standard form

  1. Are the powers the same? If not, rewrite the number with the SMALLER power so it has the bigger power: \(2\times10^{3}=0.2\times10^{4}\) (the power went up 1, so the number goes down 10 times).
  2. Add the number parts. Keep the power.
  3. Tidy up.

Small numbers? Writing both as ordinary numbers, adding, and converting back is just as good: 30 000 + 2000 = 32 000 = \(3.2\times10^{4}\).

Example Work out \((3.8\times10^{22})+(3.8\times10^{23})\).

Make both \(10^{23}\), add the numbers, keep the power.

  1. \(3.8\times10^{22}=0.38\times10^{23}\)
  2. \(0.38+3.8=4.18\)

Answer \(4.18\times10^{23}\)

Try it yourself

1
\((6\times10^{5})+(4\times10^{4})\)
Hint
\(4\times10^{4}=0.4\times10^{5}\).
Show answer

Answer \(6.4\times10^{5}\)

2
\((3\times10^{4})+(2\times10^{3})\)
Show answer

Answer \(3.2\times10^{4}\)

Subtracting

General methodHow to subtract any two numbers in standard form

  1. Make the powers the same. Tip: give the BIGGER number the smaller power, so you subtract whole numbers: \(8\times10^{9}=80\times10^{8}\).
  2. Subtract the number parts. Keep the power.
  3. Tidy up.

Example Work out \((8\times10^{9})-(9\times10^{8})\).

Write both with \(10^{8}\), subtract, then tidy.

  1. \(8\times10^{9}=80\times10^{8}\)
  2. \(80-9=71\), so \(71\times10^{8}\)
  3. \(71\times10^{8}=7.1\times10^{9}\)

Answer \(7.1\times10^{9}\)

Try it yourself

1
\((4\times10^{6})-(5\times10^{5})\)
Hint
\(4\times10^{6}=40\times10^{5}\).
Show answer

Answer \(3.5\times10^{6}\)

2
\((7\times10^{3})-(5\times10^{2})\)
Show answer

Answer \(6.5\times10^{3}\)

Squares, cubes, roots and reciprocals

General methodHow to square, cube or root a number in standard form

  1. Square / cube: do it to BOTH parts. \((A\times10^{n})^{2}=A^{2}\times10^{2n}\) (multiply the power by 2, or by 3 for a cube).
  2. Square root: the power must be EVEN. If it is odd, move one 10 into the number: \(4.9\times10^{7}=49\times10^{6}\). Then root the number and HALVE the power.
  3. Reciprocal \(\dfrac1N\): work out \(\dfrac1A\) and change the sign of the power: \(\dfrac1{4\times10^{5}}=0.25\times10^{-5}\).
  4. Tidy up.

Example Find \(\sqrt{4.9\times10^{7}}\) without a calculator.

The power 7 is odd, so make it even first, then root the number and halve the power.

  1. \(4.9\times10^{7}=49\times10^{6}\)
  2. \(\sqrt{49}=7\) and \(\sqrt{10^{6}}=10^{3}\)

Answer \(7\times10^{3}\)

Try it yourself

1
\((2\times10^{3})^{2}\)
Show answer

Answer \(4\times10^{6}\)

2
\(\sqrt{1.6\times10^{7}}\)
Hint
\(1.6\times10^{7}=16\times10^{6}\).
Show answer

Answer \(4\times10^{3}\)

3
\(\dfrac{1}{2\times10^{4}}\)
Hint
\(\tfrac12=0.5\), power \(-4\); then tidy.
Show answer

Answer \(5\times10^{-5}\)

On the calculator paper

  • Use the ×10ˣ key (or EXP): \(3.8\times10^{-4}\) is typed 3.8 ×10ˣ (−) 4. It keeps the number together.
  • Typing × 10 ^ instead? Put BRACKETS round each number when dividing: (6×10^5) ÷ (3×10^−2).
  • A display like 1.26E+25 means \(1.26\times10^{25}\): write it that way, never as \(1.26^{25}\).

Ways they test it

Type 1 Work it out, answer in standard form

"Work out (3 × 10⁴) × (5 × 10⁶). Give your answer in standard form." (2 marks)

Spot it two standard form numbers and ×, ÷, + or −.

  1. Use the method for that operation.
  2. Tidy up: the second mark is for standard form.

Type 2 Find x and y (or k)

"\((8.2\times10^{4})+(9.1\times10^{5})=x\times10^{y}\). Find \(x\) and \(y\)."

Spot it the answer is written as letters in standard form.

  1. Work out the answer in standard form.
  2. Read off: the number is \(x\), the power is \(y\).

For example \((8.2\times10^{4})+(9.1\times10^{5})=x\times10^{y}\)

  1. \(0.82\times10^{5}+9.1\times10^{5}=9.92\times10^{5}\)

Answer \(x=9.92\), \(y=5\)

Type 3 Letters in the power

"Work out \((7\times10^{a})\times(3\times10^{a+1})\) in standard form."

Spot it a letter in the power instead of a number.

  1. Same method: multiply the numbers, add the powers (as algebra).
  2. Tidy up: the power changes by 1 in the same way.

For example \((7\times10^{a})\times(3\times10^{a+1})\)

  1. \(21\times10^{2a+1}\)
  2. \(=2.1\times10^{2a+2}\)

Answer \(2.1\times10^{2a+2}\)

Type 4 Substitute into a formula

"\(a=\dfrac{b+c}{d^{2}}\). Find \(a\) when \(b=4\times10^{2}\), \(c=6\times10^{3}\), \(d=2\times10^{2}\)."

Spot it a formula and standard form values for its letters.

  1. Work out the top and the bottom separately.
  2. Divide, then tidy.

For example \(b=4\times10^{2}\), \(c=6\times10^{3}\), \(d=2\times10^{2}\)

  1. \(b+c=6400=6.4\times10^{3}\)
  2. \(d^{2}=4\times10^{4}\)
  3. \(6.4\times10^{3}\div4\times10^{4}=1.6\times10^{-1}\)

Answer \(1.6\times10^{-1}\)

Type 5 Square, root or reciprocal of a given N

"\(N=5\times10^{3}\). Find \(N^{2}\) and \(\dfrac1N\) in standard form."

Spot it one number \(N\) and a power, root or \(\tfrac1N\).

  1. Power: on both parts. Root: make the power even, halve it. Reciprocal: \(\tfrac1A\), power changes sign.
  2. Tidy up.

Type 6 Ordinary numbers in, standard form out

"Calculate \(0.3^{2}\). Give your answer in standard form."

Spot it no standard form in the question, but the answer must be.

  1. Work it out (calculator if allowed).
  2. Write the answer in standard form.

For example \(0.3^{2}\)

  1. \(0.3^{2}=0.09\)
  2. \(0.09=9\times10^{-2}\)

Answer \(9\times10^{-2}\)

Mistakes to avoid

  • ✗ \((2\times10^{3})(3\times10^{4})=6\times10^{12}\)✓ \(6\times10^{7}\)Multiplying ADDS the powers. It never multiplies them.
  • ✗ \(16.8\times10^{18}=1.68\times10^{17}\)✓ \(1.68\times10^{19}\)The number got smaller, so the power must go UP.
  • ✗ \(3\times10^{4}+2\times10^{3}=5\times10^{7}\)✓ \(3.2\times10^{4}\)Match the powers before adding.
  • ✗ \((4\times10^{4})^{2}=4\times10^{8}\)✓ \(1.6\times10^{9}\)Square the 4 too: \(16\times10^{8}\), then tidy.
  • ✗ \(16.8\times10^{18}\) left as the answer✓ \(1.68\times10^{19}\)Not standard form: you lose the last mark.

Summary

  • × : multiply the numbers, ADD the powers. ÷ : divide the numbers, SUBTRACT the powers.
  • + and − : make the powers the SAME first, then add or subtract the numbers.
  • Squares and cubes act on both parts; a square root halves an EVEN power.
  • Always finish by tidying: number ÷ 10 → power + 1; number × 10 → power − 1.
  • Calculator: use the ×10ˣ key, or brackets round each number.

If you see …

  • "Give your answer in standard form"finish with \(A\times10^{n}\), \(1\le A<10\).
  • a + or − sign between themmatch the powers first.
  • \(\sqrt{\ }\) with an odd powermove one 10 into the number first.
  • letters in the powerthe same rules, done in algebra.

Practice

Practice: easy, medium, hard

Easy

1
Work out \((2\times10^{3})\times(4\times10^{5})\).
Hint
Multiply the numbers, add the powers.
Show working
  1. \(2\times4=8\)
  2. \(10^{3+5}=10^{8}\)

Answer \(8\times10^{8}\)

2
Work out \(\dfrac{6\times10^{9}}{2\times10^{4}}\).
Hint
Divide the numbers, subtract the powers.
Show working
  1. \(6\div2=3\)
  2. \(10^{9-4}=10^{5}\)

Answer \(3\times10^{5}\)

3
Work out \((5\times10^{4})+(3\times10^{3})\).
Hint
Make both \(10^{4}\).
Show working
  1. \(3\times10^{3}=0.3\times10^{4}\)
  2. \(5+0.3=5.3\)

Answer \(5.3\times10^{4}\)

Medium

1
Work out \((6\times10^{7})\times(5\times10^{-3})\), in standard form.
Hint
\(7+(-3)=4\), and \(6\times5=30\) needs tidying.
Guide me

What is it asking? one number in standard form

What do we know? \(6\times10^{7}\) and \(5\times10^{-3}\)

Plan multiply numbers, add powers, tidy

Show working
  1. \(6\times5=30\)
  2. \(10^{7+(-3)}=10^{4}\)
  3. \(30\times10^{4}=3\times10^{5}\)

Answer \(3\times10^{5}\)

2
Work out \(\dfrac{1.2\times10^{5}}{4\times10^{8}}\), in standard form.
Hint
\(1.2\div4=0.3\).
Guide me

What is it asking? the answer in standard form

What do we know? top \(1.2\times10^{5}\), bottom \(4\times10^{8}\)

Plan divide numbers, subtract powers, then tidy 0.3

Show working
  1. \(1.2\div4=0.3\)
  2. \(10^{5-8}=10^{-3}\)
  3. \(0.3\times10^{-3}=3\times10^{-4}\)

Answer \(3\times10^{-4}\)

3
Work out \((9\times10^{6})-(2\times10^{5})\), in standard form.
Hint
\(9\times10^{6}=90\times10^{5}\).
Guide me

What is it asking? a difference in standard form

What do we know? the powers are 6 and 5

Plan match the powers, subtract, tidy

Show working
  1. \(90\times10^{5}-2\times10^{5}=88\times10^{5}\)
  2. \(=8.8\times10^{6}\)

Answer \(8.8\times10^{6}\)

Hard

1
\(N=5\times10^{3}\). Giving each answer in standard form, find (a) \(N^{2}\) (b) \(\dfrac1N\).
Hint
(b) \(\tfrac15=0.2\), so the power goes down by 1 when you tidy.
Guide me

What is it asking? \(N^{2}\) and \(\tfrac1N\), both in standard form

What do we know? \(N=5\times10^{3}\)

Plan square both parts; flip both parts; tidy each

Show working
  1. (a) \(25\times10^{6}=2.5\times10^{7}\)
  2. (b) \(\tfrac15\times10^{-3}=0.2\times10^{-3}=2\times10^{-4}\)

Answer (a) \(2.5\times10^{7}\) (b) \(2\times10^{-4}\)

PartAnswer / workingMarksGuidance
(a)\(25\times10^{6}\) seenM1
\(2.5\times10^{7}\)A1
(b)\(0.2\times10^{-3}\) seenM1
\(2\times10^{-4}\)A1
Total: 4 marks
2
\((7\times10^{a})\times(3\times10^{a-1})=k\times10^{2a}\), where \(1\le k<10\). Find \(k\).
Hint
Multiply as usual: the powers add to \(2a-1\).
Guide me

What is it asking? the number \(k\)

What do we know? \(1\le k<10\), so \(k\) is the number part of standard form

Plan multiply, add the powers, tidy so the power is 2a

Show working
  1. \(21\times10^{2a-1}\)
  2. \(=2.1\times10\times10^{2a-1}=2.1\times10^{2a}\)

Answer \(k=2.1\)

PartAnswer / workingMarksGuidance
\(21\times10^{2a-1}\)M1
\(2.1\times10^{2a}\) seenM1
\(k=2.1\)A1
Total: 3 marks
3
Find \(\sqrt{3.6\times10^{9}}\) without a calculator. Give your answer in standard form.
Hint
The power 9 is odd. Which even power can you make?
Guide me

What is it asking? a square root in standard form

What do we know? \(3.6\times10^{9}\): odd power

Plan write it as \(36\times10^{8}\), then root

Show working
  1. \(3.6\times10^{9}=36\times10^{8}\)
  2. \(\sqrt{36}\times10^{4}=6\times10^{4}\)

Answer \(6\times10^{4}\)

PartAnswer / workingMarksGuidance
\(36\times10^{8}\) seenM1
\(6\times10^{4}\)A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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