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Indices & standard form · Lesson 3 of 3

Standard form

Writing very large and very small numbers as A × 10ⁿ, multiplying and dividing in standard form, and simplifying decimals without a calculator.

14 minYou should already know: Number foundations & fractions
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A number is in standard form (scientific notation) when it’s written as

A×10n,1≤A<10, n a whole numberA \times 10^n, \qquad 1 \le A < 10,\ n \text{ a whole number}

So 345 000 000=3.45×108345\,000\,000 = 3.45 \times 10^8 and 0.000527=5.27×10−40.000527 = 5.27 \times 10^{-4}.

0.0005274 jumps right0.000527 = 5.27 × 10⁻⁴
Moving the pointEach jump right makes the number 10 times bigger, so the power of 10 goes down by 1 to keep the value

Try it

Standard form: move the pointMove the decimal point
345000000× 100
345000000 × 10⁰the same number, written another way
Each move of the point left makes the number 10 times smaller, so the power of 10 goes up by 1 to keep the value the same. Moving it right makes the power go down. Stop when exactly one non-zero digit is in front of the point.

Pick a number and move the point one place at a time. The power of 10 changes to keep the value exactly the same. Stop when there is one non-zero digit in front of the point.

More: writing numbers in standard form

Multiplying and dividing

Multiply (or divide) the numbers, and use the index laws on the powers of 10. Then tidy the result back into standard form.

(2.5×106)×(6.4×10−9)=16×10−3=1.6×10−2\begin{aligned} (2.5 \times 10^{6}) \times (6.4 \times 10^{-9}) &= 16 \times 10^{-3} \\ &= 1.6 \times 10^{-2} \end{aligned}

More: multiplying and dividing in standard form

Simplifying without a calculator

WAEC often asks you to simplify a fraction of decimals “without tables or a calculator”. Get rid of the decimals first: write each number as a whole number times a power of 10, then cancel.

Worked example · WAEC 2014

WAEC 2014 · Paper 2 · Q1 (a)

Without using tables or a calculator, simplify 0.6×32×0.0041.2×0.008×0.16\dfrac{0.6 \times 32 \times 0.004}{1.2 \times 0.008 \times 0.16}, leaving the answer in standard form (scientific notation).

  1. Remove the decimals

    Top: 0.6×32×0.004=6×10−1×32×4×10−3=768×10−40.6 \times 32 \times 0.004 = 6 \times 10^{-1} \times 32 \times 4 \times 10^{-3} = 768 \times 10^{-4}.

    Bottom: 1.2×0.008×0.16=12×10−1×8×10−3×16×10−2=1536×10−61.2 \times 0.008 \times 0.16 = 12 \times 10^{-1} \times 8 \times 10^{-3} \times 16 \times 10^{-2} = 1536 \times 10^{-6}.

    Think first. Write each decimal as a whole number times a power of 10.

  2. Divide

    768×10−41536×10−6=7681536×102=0.5×100=50\begin{aligned} \frac{768 \times 10^{-4}}{1536 \times 10^{-6}} &= \frac{768}{1536} \times 10^{2} \\ &= 0.5 \times 100 = 50 \end{aligned}

  3. Standard form

    50=5×10150 = 5 \times 10^1.

More: simplifying decimals without a calculator

Your turn

WAEC 2016 · Paper 2 · Q1 (a)

  1. (a)

    Without using mathematical tables or calculators, evaluate 0.09×1.213.3×0.00025\dfrac{0.09 \times 1.21}{3.3 \times 0.00025}, leaving the answer in standard form (scientific notation).

Worked solution (try it first)

(a)

  1. Write each decimal as a whole number times a power of 10: 9×10−2×121×10−233×10−1×25×10−5=9×12133×25×10−4−(−6)\frac{9 \times 10^{-2} \times 121 \times 10^{-2}}{33 \times 10^{-1} \times 25 \times 10^{-5}} = \frac{9 \times 121}{33 \times 25} \times 10^{-4 - (-6)}.
  2. 9×12133×25=1089825\frac{9 \times 121}{33 \times 25} = \frac{1089}{825}
    =1.32= 1.32, and 102=10010^{2} = 100, so the value is 132=1.32×102132 = 1.32 \times 10^2.

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