Unit 5 · Lesson 2 of 5
Zero and negative exponents, scientific notation
You'll be able to: rewrite negative exponents as positive ones, and work with numbers in scientific notation.
Step 1 of 5Warm-up
Two quick questions from earlier lessons. Both skills come back today. Answer from memory first.
Simplify \((2x^3)^2 \cdot x\).
Power first: \((2x^3)^2 = 2^2 x^6 = 4x^6\).
Then \(4x^6 \cdot x^1 = 4x^7\).
Answer: \(4x^7\).
Compute \(\dfrac{1}{2} - \dfrac{1}{8}\).
The LCD is 8: \(\dfrac{1}{2} = \dfrac{4}{8}\).
\(\dfrac{4}{8} - \dfrac{1}{8} = \dfrac{3}{8}\).
Answer: \(\dfrac{3}{8}\).
Step 2 of 5The idea
Watch the pattern as the exponent goes down by 1. Each time, you divide by 2:
\[2^3 = 8, \quad 2^2 = 4, \quad 2^1 = 2, \quad 2^0 = 1, \quad 2^{-1} = \tfrac{1}{2}, \quad 2^{-2} = \tfrac{1}{4}\]
So a negative exponent does not make a number negative. It means "one over": \(2^{-2} = \dfrac{1}{2^2}\). The exponent rules from last lesson still work the same way.
Zero exponent: \(x^0 = 1\) (for \(x \ne 0\)).
Negative exponent: \(x^{-n} = \dfrac{1}{x^n}\) and \(\dfrac{1}{x^{-n}} = x^n\). A factor with a negative exponent moves across the fraction bar, and its exponent becomes positive.
Only that factor moves: \(3x^{-2} = \dfrac{3}{x^2}\). The 3 has no negative exponent, so it stays.
Simplified means no zero or negative exponents are left.
Scientific notation uses this to write very big or very small numbers: \(a \times 10^n\) with \(1 \le a \lt 10\). For example, \(52{,}000 = 5.2 \times 10^4\) and \(0.0052 = 5.2 \times 10^{-3}\).
Converting: move the decimal point until one nonzero digit is in front of it. The number of places moved is the exponent. Big numbers get a positive exponent; numbers smaller than 1 get a negative one.
Multiplying and dividing: work on the front numbers, and use the exponent rules on the powers of 10. If the front number ends up outside 1 to 10, adjust it: \(15 \times 10^{-3} = 1.5 \times 10^{-2}\).
Why this shows up on the placement test: "simplify and write with positive exponents only" is one of the most common exponent questions, and scientific notation questions test the same rules.
Step 3 of 5Worked examples
Example 1: Simplify \(\dfrac{6x^{-2}y^3}{2x^4y^{-1}}\) with positive exponents
- Numbers: \(6 \div 2 = 3\).
- \(x\): quotient rule, \(x^{-2-4} = x^{-6}\).
- \(y\): \(y^{3-(-1)} = y^{3+1} = y^4\). Subtracting a negative is adding.
- So far: \(3x^{-6}y^4\). Move \(x^{-6}\) below the fraction bar: \(\dfrac{3y^4}{x^6}\).
Example 2: Scientific notation
- Write \(0.00042\) in scientific notation. Move the decimal 4 places right to get \(4.2\). The number is small, so the exponent is negative: \(4.2 \times 10^{-4}\).
- Write \(6.1 \times 10^5\) as a regular number. Move the decimal 5 places right: \(610{,}000\).
- Compute \((3 \times 10^4)(5 \times 10^{-7})\). Front numbers: \(3 \cdot 5 = 15\). Powers of 10: \(10^{4 + (-7)} = 10^{-3}\).
- \(15 \times 10^{-3}\) is not in scientific notation, since 15 is bigger than 10. \(15 = 1.5 \times 10^1\), so the answer is \(1.5 \times 10^{-2}\).
Step 4 of 5Practice
Give answers with positive exponents only. Type fractions like 3/x^4 and powers of ten like 2.5*10^3. If you miss twice, open the worked solution.
Write \(5^{-2}\) as a fraction.
Negative exponent means "one over": \(5^{-2} = \dfrac{1}{5^2}\).
Answer: \(\dfrac{1}{25}\).
Compute \(3^0 + 2^{-3}\).
\(3^0 = 1\) and \(2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}\).
\(1 + \dfrac{1}{8} = \dfrac{8}{8} + \dfrac{1}{8} = \dfrac{9}{8}\).
Answer: \(\dfrac{9}{8}\).
Simplify \(3x^{-7} \cdot x^3\). Use positive exponents only.
Product rule: \(x^{-7+3} = x^{-4}\). So you have \(3x^{-4}\).
Only the \(x^{-4}\) moves below the bar: \(\dfrac{3}{x^4}\).
Answer: \(\dfrac{3}{x^4}\).
Simplify \(\dfrac{a^{-3}b^2}{a^2b^{-4}}\). Use positive exponents only.
\(a\): \(a^{-3-2} = a^{-5}\).
\(b\): \(b^{2-(-4)} = b^6\).
\(a^{-5}b^6\) becomes \(\dfrac{b^6}{a^5}\). Answer: \(\dfrac{b^6}{a^5}\).
Simplify \((2x^{-3})^{-2}\). Use positive exponents only.
The \(-2\) goes to every factor: \(2^{-2} \cdot (x^{-3})^{-2}\).
\(2^{-2} = \dfrac{1}{4}\) and \((x^{-3})^{-2} = x^{(-3)(-2)} = x^6\).
Answer: \(\dfrac{x^6}{4}\).
In scientific notation, \(0.0000607 = 6.07 \times 10^n\). What is \(n\)?
Move the decimal right until one nonzero digit is in front: \(0.0000607 \to 6.07\). That is 5 places.
The number is smaller than 1, so the exponent is negative.
Answer: \(n = -5\).
Compute \((4 \times 10^5)(6 \times 10^{-2})\). Give the answer in scientific notation.
Front numbers: \(4 \cdot 6 = 24\). Powers of 10: \(10^{5 + (-2)} = 10^3\).
\(24 \times 10^3\): 24 is bigger than 10, so write \(24 = 2.4 \times 10^1\).
Answer: \(2.4 \times 10^4\) (which is 24,000).
Which is equal to \(\dfrac{8.4 \times 10^{-3}}{2.1 \times 10^{5}}\)?
- \(4 \times 10^{2}\)
- \(4 \times 10^{-2}\)
- \(4 \times 10^{8}\)
- \(4 \times 10^{-8}\)
Front numbers: \(8.4 \div 2.1 = 4\).
Powers of 10, quotient rule: \(10^{-3 - 5} = 10^{-8}\).
Answer: (d) \(4 \times 10^{-8}\). Choice (a) adds the exponents instead of subtracting; (b) and (c) mix up the signs or subtract in the wrong order.
Step 5 of 5Wrap-up
You can now replace \(x^0\) with 1, move negative-exponent factors across the fraction bar to make them positive, and convert, multiply, and divide numbers in scientific notation.
Read or watch next: OpenStax Intermediate Algebra 2e, 5.2 Properties of Exponents and Scientific Notation (free textbook, with more practice at the end of the section).
Also good: Khan Academy: Negative exponents (short videos). Quick rules: Exponents and polynomials cheat sheet.
Something unclear? Ask your teacher bot a follow-up question, for example: "Why is \(3x^{-2}\) equal to \(\dfrac{3}{x^2}\) and not \(\dfrac{1}{3x^2}\)?" It knows this course and can explain things another way.