Standard Form: Question 4

Worked solution

Part (a)

We need to write $5\,040\,000$ in the form $a \times 10^{n}$, where $1 \le a < 10$ and $n$ is an integer.

Step 1: Place the decimal point so that $a$ is between 1 and 10.

Starting from $5\,040\,000.0$, the decimal point must sit just after the first non-zero digit:

$5.040000\ldots \;\Rightarrow\; a = 5.04$

Step 2: Count how many places the decimal point moved.

From $5\,040\,000.$ to $5.04$ the point moves $6$ places to the left, so $n = 6$.

$\boxed{5\,040\,000 = 5.04 \times 10^{6}}$

Check: $5.04 \times 10^{6} = 5.04 \times 1\,000\,000 = 5\,040\,000$. ✓

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Part (b)

The number is $3.7 \times 10^{-4}\text{ m}$. A negative index means the number is small (less than 1), so the decimal point moves to the left.

Step: Move the decimal point 4 places to the left.

$3.7 \;\to\; 0.37 \;\to\; 0.037 \;\to\; 0.0037 \;\to\; 0.00037$

$\boxed{3.7 \times 10^{-4} = 0.00037}$

Check: $0.00037 \times 10^{4} = 0.00037 \times 10\,000 = 3.7$. ✓

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Part (c)

Work out $(6 \times 10^{8}) \times (1.5 \times 10^{-3})$.

Step 1: Multiply the number parts ($a$ values).

$6 \times 1.5 = 9$

Step 2: Multiply the powers of 10 by adding the indices.

$10^{8} \times 10^{-3} = 10^{8 + (-3)} = 10^{5}$

Step 3: Combine.

$9 \times 10^{5}$

Here $a = 9$ satisfies $1 \le a < 10$, so the answer is already in standard form.

$\boxed{(6 \times 10^{8}) \times (1.5 \times 10^{-3}) = 9 \times 10^{5}}$

Check: $6 \times 10^{8} = 600\,000\,000$ and $1.5 \times 10^{-3} = 0.0015$; then $600\,000\,000 \times 0.0015 = 900\,000 = 9 \times 10^{5}$. ✓

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Part (d)

Compare $8 \times 10^{5}$ and $2 \times 10^{6}$.

Method: compare the powers of 10 first.

For numbers in standard form, the one with the larger power of 10 is larger (provided each $a$ is between 1 and 10). Here:

$10^{6} > 10^{5}$

so $2 \times 10^{6}$ is larger, even though its $a$-value ($2$) is smaller than $8$.

$\boxed{2 \times 10^{6} \text{ is larger}}$

Check by writing as ordinary numbers:

$8 \times 10^{5} = 800\,000, \qquad 2 \times 10^{6} = 2\,000\,000$

and $2\,000\,000 > 800\,000$. ✓