Scientific Notation: Sums, Differences and Mixed Forms | Grade 8 Math | LevelKite

Before you add or subtract in scientific notation, look at what each part of the number means.

A number in scientific notation is a coefficient (the number in front) times a power of 10, such as $3 \times 10^4$ . In $10^4$ , the 10 is the base and the small raised 4 is the exponent. Every exponent in this lesson is an integer. For any nonzero number, the size of the coefficient, ignoring its sign, is at least 1 and less than 10.

Think of the power of 10 as a label that says what the coefficient counts. Since $10^4 = 10{,}000$ , the number $3 \times 10^4$ means 3 ten-thousands. Adding $3 \times 10^4$ and $2 \times 10^4$ means adding 3 ten-thousands and 2 ten-thousands. The labels match, so you add how many there are and get 5 ten-thousands. Written out, that is $30{,}000 + 20{,}000 = 50{,}000$ . In symbols:

3 \times 10^4 + 2 \times 10^4 = (3 + 2) \times 10^4 = 5 \times 10^4

Key concept

Match the powers of 10, then combine coefficients

When two numbers have the same power of 10, that power is a factor of both. The distributive property says $a \times c + b \times c = (a + b) \times c$ for any numbers $a$ , $b$ and $c$ . Let $c$ be the shared power $10^n$ , where $n$ is an integer and $a$ and $b$ are the coefficients. Then you add the coefficients and keep the power of 10. Subtraction works the same way: $a \times 10^n - b \times 10^n = (a - b) \times 10^n$ .

The power $10^n$ is defined for every integer $n$ because the base 10 is not zero.

If the powers of 10 differ, first rewrite one number so they match. At the end, check the form: a nonzero answer is in scientific notation only when the size of its coefficient is at least 1 and less than 10.

a \times 10^n + b \times 10^n = (a + b) \times 10^n

What if the powers of 10 differ, as in $(6.2 \times 10^4) + (4.5 \times 10^3)$ ? Now 6.2 counts ten-thousands and 4.5 counts thousands, so you cannot add 6.2 and 4.5 yet. First rewrite one number so both have the same power of 10, without changing its value.

Here is why you can do that. Since $4.5 = 0.45 \times 10$ , you get $4.5 \times 10^3 = 0.45 \times 10 \times 10^3 = 0.45 \times 10^4$ . The coefficient was divided by 10 and the power of 10 was multiplied by 10, so the value is still 4,500.

Move the decimal point in the coefficient one place left (divide by 10) and raise the exponent by 1 (multiply by 10).
Move it one place right (multiply by 10) and lower the exponent by 1 (divide by 10).

Either move keeps the value the same for any integer exponent, because multiplying by 10 undoes dividing by 10. You may rewrite either number. This lesson rewrites the one with the smaller exponent.

One value written with four different powers of 10

Expression	How it was rewritten	Value	In scientific notation?
$4.5 \times 10^3$	Starting form	4,500	Yes: 4.5 is at least 1 and less than 10
$0.45 \times 10^4$	Decimal point one place left, exponent raised by 1	4,500	No: 0.45 is less than 1
$0.045 \times 10^5$	Decimal point two places left, exponent raised by 2	4,500	No: 0.045 is less than 1
$45 \times 10^2$	Decimal point one place right, exponent lowered by 1	4,500	No: 45 is not less than 10

Worked example

Add when the powers of 10 differ

Find $(6.2 \times 10^4) + (4.5 \times 10^3)$ . Write the sum in scientific notation.

Compare the exponents. They are 4 and 3, so the coefficients count different place values and you cannot add them yet. Plan to rewrite the number with the smaller exponent so that both exponents are 4.
Rewrite the second number. Moving the decimal point in 4.5 one place left divides the coefficient by 10. Raising the exponent from 3 to 4 multiplies the power of 10 by 10. The two changes cancel, so the value stays 4,500.
$4.5 \times 10^3 = 0.45 \times 10^4$
Both numbers now have the same power of 10. By the distributive property, add the coefficients and keep that power.
$6.2 \times 10^4 + 0.45 \times 10^4 = (6.2 + 0.45) \times 10^4 = 6.65 \times 10^4$
Check the form: 6.65 is at least 1 and less than 10, so the answer is in scientific notation. Then check the value by writing each number out in full. The results match exactly.
$62{,}000 + 4{,}500 = 66{,}500 = 6.65 \times 10^4$

Answer

The sum is $6.65 \times 10^4$ , which is 66,500.

Worked example

Subtract small numbers and adjust the form

Find $(1.2 \times 10^{-3}) - (8.5 \times 10^{-4})$ . Write the difference in scientific notation.

Recall what a negative exponent on 10 means: divide 1 by 10 raised to the matching positive exponent. This is defined because the base 10 is not zero. So both numbers are small, and the exponent negative 3 is greater than negative 4.
$10^{-3} = \frac{1}{10^3} = 0.001 \qquad 10^{-4} = \frac{1}{10^4} = 0.0001$
Rewrite the number with the smaller exponent, negative 4. Move the decimal point in 8.5 one place left and raise the exponent by 1, to negative 3. As in the first example, the two changes cancel, so the value stays 0.00085.
$8.5 \times 10^{-4} = 0.85 \times 10^{-3}$
The powers of 10 now match, so subtract the coefficients and keep that power.
$1.2 \times 10^{-3} - 0.85 \times 10^{-3} = (1.2 - 0.85) \times 10^{-3} = 0.35 \times 10^{-3}$
Check the form. The coefficient 0.35 is less than 1, so this is not yet scientific notation. Move the decimal point one place right, which multiplies the coefficient by 10, and lower the exponent by 1, to negative 4, so the value does not change.
$0.35 \times 10^{-3} = 3.5 \times 10^{-4}$
Now 3.5 is at least 1 and less than 10. Check the value by writing each number out in full.
$0.0012 - 0.00085 = 0.00035 = 3.5 \times 10^{-4}$

Answer

The difference is $3.5 \times 10^{-4}$ , which is 0.00035.

Sometimes one number is in scientific notation and the other is in decimal notation, the everyday place-value way of writing a number, such as 80,000 or 0.0031. The two notations are different ways to write a value, so you can switch a number from one to the other without changing it.

The method in this lesson needs both numbers in scientific notation. So when the forms are mixed, make this your first move: rewrite the decimal number in scientific notation. Then add or subtract as before: match the powers of 10, combine the coefficients and check the form.

You can check your result by writing both numbers in decimal notation instead. Both routes give the same value.

Worked example

Add numbers written in two different forms

Find $(9.6 \times 10^5) + 80{,}000$ . Write the sum in scientific notation.

The numbers are in different forms, so write 80,000 in scientific notation. Move the decimal point 4 places left to get the coefficient 8. That divides by 10 four times, so multiply by 10 four times to keep the value: use the exponent 4.
$80{,}000 = 8 \times 10^4$
The exponents are 5 and 4, so the powers of 10 do not match yet. Rewrite the number with the smaller exponent: decimal point one place left, exponent up by 1.
$8 \times 10^4 = 0.8 \times 10^5$
The powers of 10 match, so add the coefficients and keep that power.
$9.6 \times 10^5 + 0.8 \times 10^5 = (9.6 + 0.8) \times 10^5 = 10.4 \times 10^5$
Check the form. The coefficient 10.4 is not less than 10. Move the decimal point one place left, which divides the coefficient by 10, and raise the exponent by 1 so the value does not change.
$10.4 \times 10^5 = 1.04 \times 10^6$
Now 1.04 is at least 1 and less than 10. Check the value in decimal notation.
$960{,}000 + 80{,}000 = 1{,}040{,}000 = 1.04 \times 10^6$

Answer

The sum is $1.04 \times 10^6$ , which is 1,040,000.

Three mixed-form problems: rewrite, then operate

Problem	Both in scientific notation	Work	Result
$(7.5 \times 10^6) - 300{,}000$	$(7.5 \times 10^6) - (3 \times 10^5)$	$7.5 \times 10^6 - 0.3 \times 10^6 = (7.5 - 0.3) \times 10^6$	$7.2 \times 10^6$
$0.0031 + (4 \times 10^{-4})$	$(3.1 \times 10^{-3}) + (4 \times 10^{-4})$	$3.1 \times 10^{-3} + 0.4 \times 10^{-3} = (3.1 + 0.4) \times 10^{-3}$	$3.5 \times 10^{-3}$
$(8.4 \times 10^6) \div 2{,}000$	$(8.4 \times 10^6) \div (2 \times 10^3)$	$(8.4 \div 2) \times 10^{6-3}$	$4.2 \times 10^3$

The big idea: the power of 10 is a label, and you can add or subtract coefficients only when they share the same label. So make the powers of 10 match, combine the coefficients and keep that power. Then check that a nonzero answer has a coefficient whose size is at least 1 and less than 10. If one number is in decimal notation, rewrite it in scientific notation first.

Next, you will use these skills to solve word problems and to find and correct errors in operations with scientific notation.