Multiplying and Dividing in Scientific Notation | Grade 8 Math | LevelKite

You already know that a nonzero number is in scientific notation when it is written as $a \times 10^n$ . Here $n$ is an integer, and the coefficient $a$ has a size (absolute value) of at least $1$ and less than $10$ : $1 \le |a| < 10$ . For example, $4{,}500{,}000 = 4.5 \times 10^6$ and $0.03 = 3 \times 10^{-2}$ .

In this lesson you will multiply and divide numbers written in scientific notation. You will also write each answer in scientific notation.

A number in scientific notation is a product of two factors: its coefficient and a power of ten. So when you multiply two such numbers, you multiply four factors. You can multiply factors in any order and group them in any way. That means you can put the two coefficients together and the two powers of ten together.

Both powers have the same base, $10$ , so you multiply them by adding their exponents. For example, $10^3$ is three factors of $10$ and $10^5$ is five more factors. So $10^3 \times 10^5 = 10^{3+5} = 10^8$ . The rule $10^m \times 10^n = 10^{m+n}$ holds for all integers $m$ and $n$ .

Negative exponents follow the same rule. A negative exponent means dividing by factors of ten, which is allowed because $10$ is not zero. So $10^5 \times 10^{-3} = 10^5 \div 10^3 = 10^2$ , which matches $5 + (-3) = 2$ .

Here is a complete product worked this way:

(2 \times 10^3)(4 \times 10^5) = (2 \times 4) \times (10^3 \times 10^5) = 8 \times 10^{3+5} = 8 \times 10^8

Worked example

Multiplying when the coefficient reaches 10 or more

Multiply $(4.5 \times 10^6)(3 \times 10^{-2})$ . Write the answer in scientific notation.

Put the coefficients together and the powers of ten together. The value does not change, because you can multiply factors in any order and with any grouping.
$(4.5 \times 10^6)(3 \times 10^{-2}) = (4.5 \times 3) \times (10^6 \times 10^{-2})$
Multiply the coefficients. Think of 4.5 as 4 plus 0.5, and multiply each part by 3.
$4.5 \times 3 = 4 \times 3 + 0.5 \times 3 = 12 + 1.5 = 13.5$
Multiply the powers of ten by adding their exponents. You can do this because both powers have the same base, 10.
$10^6 \times 10^{-2} = 10^{6+(-2)} = 10^4$
The result so far is 13.5 times 10 to the 4th power. Its value is correct, but the coefficient 13.5 is not less than 10. So it is not in scientific notation yet. Rewrite 13.5 as 1.35 times 10. The value stays the same, because 1.35 × 10 = 13.5.
$13.5 \times 10^4 = (1.35 \times 10^1) \times 10^4$
Group the powers of ten and add their exponents. The new coefficient, 1.35, is at least 1 and less than 10, so the answer is now in scientific notation.
$1.35 \times (10^1 \times 10^4) = 1.35 \times 10^{1+4} = 1.35 \times 10^5$
Check the answer with decimal notation. The first number is 4,500,000 and the second number is 0.03. Their product matches the answer.
$4{,}500{,}000 \times 0.03 = 135{,}000 = 1.35 \times 10^5$

Answer

The product is $1.35 \times 10^5$ , which equals $135{,}000$ .

Dividing works the same way. Write the quotient as a fraction. Then split it into a fraction of coefficients times a fraction of powers of ten. This is valid because when you multiply two fractions, you multiply their tops and multiply their bottoms. The divisor must not be zero. A number in scientific notation is never zero, because its coefficient has size at least $1$ .

Divide the powers of ten by subtracting exponents, top minus bottom: $10^m \div 10^n = 10^{m-n}$ for all integers $m$ and $n$ . This is true because $10^{m-n} \times 10^n = 10^{(m-n)+n} = 10^m$ . So $10^{m-n}$ is the number you multiply $10^n$ by to get $10^m$ . The division is always allowed, because the base, $10$ , is not zero, so $10^n$ is never zero. For example:

\frac{8 \times 10^9}{2 \times 10^3} = \frac{8}{2} \times \frac{10^9}{10^3} = 4 \times 10^{9-3} = 4 \times 10^6

Key concept

Work on each part, then fix the coefficient

To multiply, multiply the coefficients and add the exponents. To divide, divide the coefficients and subtract the exponents, top minus bottom. Here $a$ and $b$ are coefficients with $1 \le |a| < 10$ and $1 \le |b| < 10$ , so $b \ne 0$ . Also, $m$ and $n$ are integers.

Then check the new coefficient. A product $ab$ has size at least $1$ and less than $100$ . A quotient $a \div b$ has size greater than $0.1$ and less than $10$ . So you need at most one fix, and each fix keeps the value the same:

If the size is $10$ or more, divide the coefficient by $10$ and add $1$ to the exponent.
If the size is less than $1$ , multiply the coefficient by $10$ and subtract $1$ from the exponent.

(a \times 10^m)(b \times 10^n) = (ab) \times 10^{m+n}, \qquad \frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}

Fixing the coefficient after multiplying or dividing

Problem	Result before fixing	Check the size, then rewrite the coefficient	Combine the powers of ten
$(4 \times 10^2)(6 \times 10^3)$	$24 \times 10^5$	$24$ is $10$ or more: $24 = 2.4 \times 10^1$	$2.4 \times 10^{1+5} = 2.4 \times 10^6$
$(2 \times 10^{-1})(5 \times 10^4)$	$10 \times 10^3$	$10$ is not less than $10$ : $10 = 1 \times 10^1$	$1 \times 10^{1+3} = 1 \times 10^4$
$(3 \times 10^5) \div (5 \times 10^7)$	$0.6 \times 10^{-2}$	$0.6$ is less than $1$ : $0.6 = 6 \times 10^{-1}$	$6 \times 10^{-1+(-2)} = 6 \times 10^{-3}$
$(-1 \times 10^6) \div (4 \times 10^{-3})$	$-0.25 \times 10^9$	The size, $0.25$ , is less than $1$ : $-0.25 = -2.5 \times 10^{-1}$	$-2.5 \times 10^{-1+9} = -2.5 \times 10^8$

Worked example

Dividing measurements: how long sunlight takes

The sun is about $1.5 \times 10^{11}$ meters from Earth, and light travels about $3 \times 10^8$ meters per second. About how many seconds does sunlight take to reach Earth? Give the answer in scientific notation.

Let t be the travel time in seconds. Time equals distance divided by speed. Meters divided by meters per second gives seconds. Both measurements are rounded, so t is also approximate.
$t \approx \frac{1.5 \times 10^{11}\ \text{m}}{3 \times 10^{8}\ \text{m/s}}$
Split the fraction into coefficients and powers of ten, as before. Divide the coefficients. Then subtract the exponents, top minus bottom.
$\frac{1.5}{3} \times \frac{10^{11}}{10^{8}} = 0.5 \times 10^{11-8} = 0.5 \times 10^{3}$
The coefficient 0.5 is less than 1. Rewrite 0.5 as 5 times one-tenth, which has the same value. Then add the exponents.
$0.5 \times 10^{3} = (5 \times 10^{-1}) \times 10^{3} = 5 \times 10^{-1+3} = 5 \times 10^{2}$
To get a sense of this time, write it as 500 seconds. Then divide by 60, the number of seconds in a minute.
$5 \times 10^{2}\ \text{s} = 500\ \text{s} \approx 8.3\ \text{min}$

Answer

Sunlight takes about $5 \times 10^2$ seconds, or about $500$ seconds, to reach Earth. That is a little more than $8$ minutes.

Worked example

Dividing with a negative coefficient and negative exponents

Find $(-7.2 \times 10^{-3}) \div (9 \times 10^{-7})$ . Write the answer in scientific notation.

Write the quotient as a fraction and split it into coefficients and powers of ten. The divisor is not zero, so the quotient is defined.
$\frac{-7.2 \times 10^{-3}}{9 \times 10^{-7}} = \frac{-7.2}{9} \times \frac{10^{-3}}{10^{-7}}$
Divide the coefficients. Because 9 × 0.8 = 7.2, you know that 7.2 ÷ 9 = 0.8. A negative number divided by a positive number is negative.
$\frac{-7.2}{9} = -0.8$
Subtract the exponents, top minus bottom. Subtracting negative 7 is the same as adding 7.
$\frac{10^{-3}}{10^{-7}} = 10^{-3-(-7)} = 10^{-3+7} = 10^{4}$
The size of the coefficient is 0.8, which is less than 1. Rewrite negative 0.8 as negative 8 times one-tenth, then add the exponents. The negative sign belongs to the coefficient. The base of the power is still 10.
$-0.8 \times 10^{4} = (-8 \times 10^{-1}) \times 10^{4} = -8 \times 10^{-1+4} = -8 \times 10^{3}$

Answer

The quotient is $-8 \times 10^3$ , which equals $-8{,}000$ .

Takeaway: To multiply or divide numbers in scientific notation, work with the coefficients and the powers of ten separately. Multiply or divide the coefficients, and add or subtract the exponents. Then make the coefficient's size at least $1$ and less than $10$ . Change the exponent so the value stays the same.

Next, you will add and subtract numbers written in scientific notation. You will also work with numbers given in both decimal and scientific notation.