Topic 1 of 8

Exponent rules and scientific notation

Step 10 · Pre-algebra topics

Use the rules for multiplying, dividing and raising powers, make sense of zero and negative exponents, and write very large and very small numbers in scientific notation.

Exponent rules

An exponent says how many times the base is multiplied by itself: 2⁵ means 2 × 2 × 2 × 2 × 2 = 32.

Counting the factors gives three rules for powers with the same base.

Rule	Example	Why
To multiply, add the exponents	x³ · x⁴ = x⁷	3 factors and 4 more make 7
To divide, subtract the exponents	x⁶ ÷ x² = x⁴	2 factors cancel
For a power of a power, multiply	(x²)³ = x⁶	3 groups of 2 factors

Example: Using the rules

Simplify 3⁴ · 3² and 5⁸ ÷ 5⁵.

3⁴ · 3² is 3⁶: add the exponents.
5⁸ ÷ 5⁵ is 5³, which is 125.

Answer3⁶ and 125

Example: A product to a power

Simplify (2x³)².

Square each factor: 2² · (x³)².
That is 4 · x⁶.

Answer4x⁶

Zero and negative exponents

x³ ÷ x³ is 1, and the subtraction rule says it is x⁰. So any number other than 0, raised to the power 0, is 1.

Keep dividing by the base and the exponent goes below zero. A negative exponent means one over the power.

Power	Value
2³	8
2²	4
2¹	2
2⁰	1
2⁻¹	$\frac{1}{2}$
2⁻²	$\frac{1}{4}$

a^{-n} = \frac{1}{a^{n}}

Example: Negative exponents

Write 10⁻³ as a fraction and as a decimal.

10⁻³ is $\frac{1}{10^{3}}$ , which is $\frac{1}{1000}$ .
As a decimal that is 0.001.

Answer $\frac{1}{1000}$ , or 0.001

Scientific notation

Scientific notation writes a number as a number from 1 up to (but not including) 10, times a power of 10. Very big and very small numbers become easy to read and compare.

Number	Scientific notation
4,500,000	4.5 × 10⁶
93,000,000	9.3 × 10⁷
0.0007	7 × 10⁻⁴
0.000052	5.2 × 10⁻⁵

Example: Writing a big number

Write 380,000 in scientific notation.

Put the point after the first digit: 3.8.
The point moved 5 places, so the power is 10⁵.

Answer3.8 × 10⁵

Example: Multiplying

What is (3 × 10⁴) × (2 × 10⁵)?

Multiply the numbers: 3 × 2 = 6.
Multiply the powers: 10⁴ × 10⁵ is 10⁹.

Answer6 × 10⁹

Example: Comparing

Which is larger: 7 × 10⁻³ or 2 × 10⁻²?

Compare the powers first: 10⁻² is ten times 10⁻³.
As decimals they are 0.007 and 0.02.

Answer2 × 10⁻²

Try it yourself

Work each one on paper first, then open the answer to check.

Simplify 4² · 4³ as a single power.
Show answer
4⁵
Simplify (y⁴)³.
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y¹²
What is 7⁰?
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1
Write 2⁻³ as a fraction.
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$\frac{1}{8}$
Write 0.00062 in scientific notation.
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6.2 × 10⁻⁴
Write 5.1 × 10⁶ as an ordinary number.
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5,100,000
What is (4 × 10³) × (2.5 × 10²)?
Show answer
4 × 2.5 = 10, so it is 10 × 10⁵, which is 1 × 10⁶.