Standard Form: Question 1

Worked solution

Part (a): write the mass in standard form

A number in standard form is $a \times 10^{n}$, where $1 \le a < 10$ and $n$ is an integer.

Start from $0.000021$ and move the decimal point to just after the first non-zero digit (the $2$):

$0.000021 = 2.1 \times 10^{-5}.$

The decimal point moved 5 places to the right, and because the original number is less than 1 the index is negative. So

$0.000021\ \text{kg} = 2.1 \times 10^{-5}\ \text{kg}.$

Part (b): total mass of the sack

Method 1: multiply first, then convert.

$\text{total mass} = 0.000021 \times 30\,000 = 0.63\ \text{kg}.$

Now convert $0.63$ to standard form:

$0.63 = 6.3 \times 10^{-1}\ \text{kg}.$

Method 2: work in standard form throughout, using the index laws.

Write both numbers in standard form first:

$0.000021 = 2.1 \times 10^{-5}, \qquad 30\,000 = 3 \times 10^{4}.$

Multiply the coefficients and add the indices, since $10^{a} \times 10^{b} = 10^{a+b}$:

$(2.1 \times 3) \times 10^{-5+4} = 6.3 \times 10^{-1}\ \text{kg}.$

Both methods give the same answer:

$\boxed{6.3 \times 10^{-1}\ \text{kg}}.$

Check: the coefficient $6.3$ satisfies $1 \le 6.3 < 10$, so the answer is correctly in standard form.