 What you need to know

Things to remember:

• A positive power of 10 means how many times to multiply by 10.
• Multiplying by 10 means we move the digits left, so for each power of 10 the digits move left one place.
• We move the digits left as many times as the power.
• A negative power of 10 means how many times to divide by 10.
• Dividing by 10 means we move the digits right, so for each power of 10 the digits move right one place.
• We move the digits right as many times as the power.

Standard form is typically used to for extremely large or extremely small numbers, such as measuring the speed of light or the size of bacteria. Start form numbers are written like this:

$$a\times10^n$$

Where $a$ is any number between 1 and 10, but not including 10, and $n$ is a positive or negative integer.

We can turn numbers in standard form into ordinary numbers quite easily, and we’ll look at it in two steps for positive powers and 2 steps for negative powers.

Write $3.2\times10^4$

Step 1: Think about what the power of 10 means.

Our power of 10 just means that we multiply by 10 4 times.

$$10^4=10\times 10\times 10\times 10$$

Step 2: Instead of a power of 10, write as multiply by 10.

$$3.2\times10^4 =3.2\times10\times 10\times 10\times 10$$

Step 3: Do the multiplication by 10 one at a time.

$$3.2\times10^4 =32\times 10\times 10\times 10$$

$$3.2\times10^4 =320\times 10\times 10$$

$$3.2\times10^4 =3200\times 10$$

$$3.2\times10^4 =32000$$

Notice how the number of times we moved the digits left was the same number as the power. This is the true for any positive power.

Evaluate $8.3\times10^5$

$$8.3\times10^5=8.3\times10 \times10 \times10 \times10 \times10$$

$$8.3\times10^5=83\times10 \times10 \times10 \times10$$

$$8.3\times10^5=830 \times10 \times10 \times10$$

$$8.3\times10^5=8300 \times10 \times10$$

$$8.3\times10^5=83000 \times10$$

$$8.3\times10^5=830000$$

Evaluate $2.17\times10^6$

We’re going to look at this one in a slightly different way. We’re going to keep it in standard form, but minus one from the power each time we move the digits left.

$$2.17\times10^6$$

$$21.7\times10^5$$

$$217\times10^4$$

Now, we know that if we are multiply by 10 4 times, this means we will just add 4 zeros.

$$217\times10^4 =2170000$$

So, this gives us a quick way of doing our questions.

Step 1: Subtract how many decimal places there are from the power.

Step 2: Remove the decimal point, add as many zeros as the number in Step 1.

Evaluate $3.298\times10^8$

Step 1: Subtract how many decimal places there are from the power.

There are 3 decimal places.

The power is 8.

$$8-3=5$$

Step 2: Remove the decimal point, add as many zeros as the number in Step 1.

$$3.298\times10^8=329800000$$

Our values of $n$ in $a\times10^n$ can be negative, but we just need to remember that negative exponents mean that we divide instead of multiply.

Evaluate $5.13\times10^{-3}$

$$5.13\times10^{-3}=5.13\div10\div10\div10$$

$$5.13\times10^{-3}=0.513\div10\div10$$

$$5.13\times10^{-3}=0.0513\div10$$

$$5.13\times10^{-3}=0.00513$$

If we look at this answer, we might be able to notice something nice. The number of 0s on the left of our number, including the one before the decimal, is the same the positive value of our power. This gives us a quick way of writing these.

Step 2: Write as many 0s, including one before the decimal, and then write your number.

Evaluate $9.89\times10^{-4}$

4

Step 2: Write as many 0s, including one before the decimal, and then write your number.

0.000989

Example Questions

Step 1: Subtract how many decimal places there are from the power.

There are 2 decimal places.

The power is 6.

$$6-2=4$$

Step 2: Remove the decimal point, add as many zeros as the number in Step 1.

$$7.23\times10^6=7230000$$

3

Step 2: Write as many 0s, including one before the decimal, and then write your number.

0.00123