Number

YOU ARE LEARNING:

Standard Form: Adding and Subtracting

We can also add and subtract numbers in standard form without having to convert them to normal numbers.

When we are adding and subtracting really large numbers, it's actually easier when they are in standard form!

We can only add and subtract numbers in standard form if the powers of 10 are the same.

This is because we want the final answer to be in standard form, where a is between $1$ and $10$ and with just one power of 10.

$a \times 10^x$

Choose an expression that could be added with the following expression:

$4.2 \times 10^7$

So how do we change the power of a number in standard form?

Write $2.4 \times 10^6$ in terms of $10^5$ .

We need to divide the power by $10$ .

$10^6 \div 10 = 10^5$

To keep the number equivalent, we need to counteract the fact we have one less power of 10.

This means the value of $a$ needs to have one more power of 10.

Multiply $2.4 \times 10$

Put the two parts together

$2.4 \times 10^6 \rightarrow 24 \times 10^5$

Let's try another example:

Express $5.6 \times 10^3$ in terms of $10^5$

We need to multiply the powers by $10^2$ ( $100$ ).

$10^3 \times 10^2 = 10^5$

Counteract the two extra powers of 10

To make sure our answer is still equivalent to the original number, we divide $a$ by $10^2$ .

Work out $5.6 \div 10^2$

Now put the two parts together.

Which of the following is equivalent to $3.1 \times 10^4$ ?

Write $1.3 \times 10^6$ in terms of $10^4$

Now let's look at an example of adding two numbers in standard form:

Notice that the powers of 10 are different here

We can't add the numbers together until we have equalised the powers of 10.

First, make $2.5 \times 10^4$ in terms of $\times 10^5$

As the power needs to increase by 10 times, we need to divide the number by 10: $2.5 \times 10^4=0.25\times 10^5$

The powers are now the same

The powers of 10 are now the same, so our calculation is $(0.25 \times 10^5)+(2.5 \times 10^5)$ .To add these together, we just have to add the decimals together.

Add the decimals

$2.5+0.25=2.75$ , which leaves us with $2.75\times10^5$ .

Nice!

We've added the numbers together, without needing to convert them back to normal numbers.

Let's try another, calculate and give your answer in correct standard form:

First, we must make the powers the same.

We will try this question by writing both numbers in terms of $10^2$ .

Express $(9.2 \times 10^4)$ in terms of $10^2$ .

Now add the two number parts together: $8.2 + 920$

Putting the number with index we have: $928.2 \times 10^2$

However, this is not in correct standard form!

We need to make the value of $a$ is less than $10$ .

We will have to divide $928.2 \div 100 = 9.282$

This means the index will need two extra powers of 10.

Final answer will be $9.282 \times 10^4$

To 3 significant figures, find:

$(2.67 \times 10^8) + (2.12 \times 10^6)$

Let's try subtracting! To 3 significant figures, find:

$(3.45 \times 10^5) - (5.12 \times 10^4)$

Are the powers the same here? Yes or No

They aren't, so we need to equalise them

Let's make $5.12\times10^4$ in terms of $10^5$ . In order to do this, we will need to divide the decimal by 10.

What is $5.12 \times 10^4$ in terms of $10^5$ ?

Now we can subtract the decimals

$3.45 - 0.512 = 2.938$

As the question asked for 3 significant figures, round $2.938$ to 3sf.

So the final answer is $2.94\times 10^5$

Awesome! 😎

Subtract $(8 \times 10^7) - (3.5 \times 10^6)$