Lesson 46: Solving Problems Using Scientific Notation

New Concept

Learn to multiply and divide with scientific notation by handling coefficients and exponents separately, then adjusting the result to its proper form.

Multiplication: To multiply numbers written in scientific notation, we multiply the coefficients to find the coefficient of the product. Then we multiply the powers of 10 by adding the exponents.

Division: To divide numbers in scientific notation we divide the coefficients, and we divide the powers of 10 by subtracting the exponents.

What’s next

This card lays the groundwork. Next, you'll tackle worked examples and solve real-world problems involving light-years and astronomical distances.

Property

To multiply numbers in scientific notation, multiply the coefficients and add the exponents. $(a \times 10^m)(b \times 10^n) = (a \times b) \times 10^{m+n}$

Examples

$(1.2 \times 10^5)(3.0 \times 10^5) = 3.6 \times 10^{10}$
$(7.5 \times 10^3)(2.0 \times 10^5) = 15.0 \times 10^8 = 1.5 \times 10^9$
$(4.0 \times 10^6)(5.0 \times 10^3) = 20.0 \times 10^9 = 2.0 \times 10^{10}$

Explanation

Think of it as a team effort! The front numbers (coefficients) multiply together, while the powers of 10 team up by adding their exponents. If the lead number gets too big (10 or more), just slide the decimal and give the exponent a boost to fix it.

Property

To divide numbers in scientific notation, divide the coefficients and subtract the exponents. $\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}$

Examples

$\frac{9.6 \times 10^7}{3.0 \times 10^2} = \frac{9.6}{3.0} \times 10^{7-2} = 3.2 \times 10^5$
$\frac{3.0 \times 10^8}{4.0 \times 10^4} = 0.75 \times 10^4 = 7.5 \times 10^3$
$\frac{7.5 \times 10^6}{2.5 \times 10^6} = 3 \times 10^0 = 3$

Explanation

It's the reverse of multiplication! Divide the front numbers and subtract the exponents of the powers of 10. If your final coefficient is less than 1, you'll need to slide the decimal point and lower the exponent to get it back into the right format.

Property

In proper scientific notation, the coefficient must be a number greater than or equal to 1 and less than 10 ($1 \le |c| < 10$).

Examples

• $15.0 \times 10^8 \rightarrow$ Move decimal left 1 spot, add 1 to exponent $\rightarrow 1.5 \times 10^9$
• $0.75 \times 10^4 \rightarrow$ Move decimal right 1 spot, subtract 1 from exponent $\rightarrow 7.5 \times 10^3$
• $345 \times 10^2 \rightarrow$ Move decimal left 2 spots, add 2 to exponent $\rightarrow 3.45 \times 10^4$

Explanation

The coefficient has a strict rule: be a number from 1 up to, but not including, 10. If your result is a rule-breaker like 15.0 or 0.25, move the decimal! For every spot you move the decimal, adjust the exponent on the 10 to keep the value balanced.