A single grain of rice is tiny, yet a chessboard story says that doubling one grain on each square would pile up a mountain of rice. Powers are the shortcut that makes such huge (and microscopic) numbers easy to write and compare. In this chapter you will master the rules of exponents, extend them to zero and negative exponents, and use scientific notation to handle the biggest and smallest quantities in science.
1. What is an exponent?
For a number \(a\) and a positive integer \(n\), the power \(a^n\) means \(a\) multiplied by itself \(n\) times:
\[ a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ factors}} \]
The number \(a\) is the base and \(n\) is the exponent. We also say that \(a^n\) is “\(a\) to the \(n\)th power”.
For example, \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\) and \(10^3 = 10 \times 10 \times 10 = 1{,}000\). By convention, \(a^1 = a\). The exponent only counts the factors: \(2^5\) is not \(2 \times 5\).
The chart shows how quickly powers grow: each time the exponent goes up by 1, the value of \(2^n\) doubles. Powers of 10 are even more dramatic, and they are the key to scientific notation later in this chapter.
The exponent applies only to what is directly in front of it. \((-3)^2 = (-3)(-3) = 9\), but \(-3^2 = -(3 \times 3) = -9\). An odd exponent keeps a negative base negative: \((-2)^3 = -8\).
2. The product rule
What happens when you multiply two powers with the same base? Write them out: \(3^2 \times 3^4 = (3 \times 3) \times (3 \times 3 \times 3 \times 3) = 3^6\). There are \(2 + 4 = 6\) factors in total.
For any nonzero base \(a\) and integers \(m\) and \(n\):
\[ a^m \times a^n = a^{m+n} \]
When multiplying powers with the same base, keep the base and add the exponents.
Simplify \(5^3 \times 5^4\) and \((4x^2)(3x^5)\).
\(5^3 \times 5^4 = 5^{3+4} = 5^7\).
Multiply the numbers, then use the rule on the variable: \((4x^2)(3x^5) = 12x^{2+5} = 12x^7\).
The rule works only when the bases match. \(2^3 \times 3^2 = 8 \times 9 = 72\) cannot be written as one power of 6. And never multiply the bases: \(2^3 \times 2^4 = 2^7\), not \(4^7\).
3. The quotient rule
Dividing powers cancels common factors. For instance, \(\dfrac{7^5}{7^2} = \dfrac{7 \times 7 \times 7 \times 7 \times 7}{7 \times 7} = 7 \times 7 \times 7 = 7^3\).
For any nonzero base \(a\) and integers \(m\) and \(n\):
\[ \frac{a^m}{a^n} = a^{m-n} \]
When dividing powers with the same base, keep the base and subtract the exponents.
Simplify \(\dfrac{10^9}{10^4}\) and \(\dfrac{12a^8}{4a^3}\).
\(\dfrac{10^9}{10^4} = 10^{9-4} = 10^5 = 100{,}000\).
Divide the numbers and use the rule on the variable: \(\dfrac{12a^8}{4a^3} = 3a^{8-3} = 3a^5\).
4. Power of a power
Now raise a power to another power: \((2^3)^2 = 2^3 \times 2^3 = 2^{3+3} = 2^6 = 64\). Two groups of three factors make six factors.
For nonzero \(a\) and \(b\) and integers \(m\) and \(n\):
\[ (a^m)^n = a^{m \times n} \qquad (ab)^n = a^n b^n \qquad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \]
When raising a power to a power, multiply the exponents. An exponent outside parentheses applies to every factor inside.
Simplify \((x^4)^5\) and \((3x^2)^3\).
\((x^4)^5 = x^{4 \times 5} = x^{20}\).
\((3x^2)^3 = 3^3 \times (x^2)^3 = 27x^6\). The exponent 3 hits both the 3 and the \(x^2\).
Compare \((2^3)^2 = 2^6 = 64\) with \(2^3 \times 2^2 = 2^5 = 32\). Powers multiplied together: add exponents. Power raised to a power: multiply exponents.
5. Zero and negative exponents
What should \(2^0\) or \(2^{-3}\) mean? Look for a pattern. Each time the exponent drops by 1, the value is divided by 2:
| Power | Value |
|---|---|
| \(2^3\) | 8 |
| \(2^2\) | 4 |
| \(2^1\) | 2 |
| \(2^0\) | 1 |
| \(2^{-1}\) | \(\dfrac{1}{2}\) |
| \(2^{-2}\) | \(\dfrac{1}{4}\) |
| \(2^{-3}\) | \(\dfrac{1}{8}\) |
To keep the pattern (and the rules above) working, we define the following.
For any nonzero number \(a\) and positive integer \(n\):
\[ a^0 = 1 \qquad\qquad a^{-n} = \frac{1}{a^n} \]
A negative exponent means “take the reciprocal”. It does not make the number negative.
Evaluate \(5^{-2}\) and \(\left(\dfrac{2}{3}\right)^{-2}\), then simplify \(x^3 \times x^{-5}\).
\(5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}\).
\(\left(\dfrac{2}{3}\right)^{-2} = \left(\dfrac{3}{2}\right)^2 = \dfrac{9}{4}\), because a negative exponent flips the fraction.
\(x^3 \times x^{-5} = x^{3+(-5)} = x^{-2} = \dfrac{1}{x^2}\).
On my planet we say: “negative exponent, flip it upstairs or downstairs.” Move the power to the other side of the fraction bar and its exponent turns positive!
\(2^{-3}\) equals \(\dfrac{1}{8}\), not \(-8\). Also remember \(a^0 = 1\), never 0 (for \(a \neq 0\)).
6. Scientific notation
Scientists deal with numbers like the number of seconds in a century or the width of a virus. Writing all those zeros is slow and error-prone, so we use powers of 10.
A number is in scientific notation when it is written as
\[ a \times 10^n \qquad \text{with } 1 \le |a| < 10 \text{ and } n \text{ an integer.} \]
The factor \(a\) has exactly one nonzero digit to the left of the decimal point.
- Place the decimal point after the first nonzero digit to get \(a\).
- Count how many places the decimal point moved.
- If you moved it to the left (large number), the exponent is positive. If you moved it to the right (small number), the exponent is negative.
Write \(45{,}300{,}000\) and \(0.00062\) in scientific notation, then write \(3.8 \times 10^5\) in standard form.
\(45{,}300{,}000 = 4.53 \times 10^7\) (7 places to the left).
\(0.00062 = 6.2 \times 10^{-4}\) (4 places to the right).
\(3.8 \times 10^5 = 380{,}000\) (move the decimal point 5 places to the right).
\(45 \times 10^6\) and \(0.8 \times 10^3\) are correct values but not in scientific notation, because \(a\) is not between 1 and 10. Rewrite them as \(4.5 \times 10^7\) and \(8 \times 10^2\).
7. Operations with scientific notation
Multiplying and dividing. Group the decimal parts and the powers of 10 separately, use the product or quotient rule on the powers of 10, and finally fix the decimal part if it leaves the range from 1 to 10.
Compute \((3 \times 10^4)(5 \times 10^6)\) and \(\dfrac{8.4 \times 10^9}{2.1 \times 10^4}\).
\((3 \times 10^4)(5 \times 10^6) = 15 \times 10^{10} = 1.5 \times 10^{11}\).
\(\dfrac{8.4 \times 10^9}{2.1 \times 10^4} = \dfrac{8.4}{2.1} \times 10^{9-4} = 4 \times 10^5\).
Adding and subtracting. You can only add or subtract terms that carry the same power of 10. If the powers differ, rewrite one number first.
Compute \(3.2 \times 10^5 + 4.1 \times 10^5\) and \(6 \times 10^4 + 5 \times 10^3\).
Same power: \(3.2 \times 10^5 + 4.1 \times 10^5 = (3.2 + 4.1) \times 10^5 = 7.3 \times 10^5\).
Different powers: \(5 \times 10^3 = 0.5 \times 10^4\), so \(6 \times 10^4 + 0.5 \times 10^4 = 6.5 \times 10^4\).
8. Comparing very large and small quantities
Scientific notation makes comparisons easy. For positive numbers, first compare the exponents: the larger exponent wins. If the exponents are equal, compare the decimal parts \(a\). To find how many times larger one quantity is than another, divide them.
The Moon is about \(3.84 \times 10^8\) m from Earth, and the Sun is about \(1.5 \times 10^{11}\) m away. Which is farther, and by what factor?
Since \(11 > 8\), the Sun is farther. Divide: \(\dfrac{1.5 \times 10^{11}}{3.84 \times 10^8} = \dfrac{1.5}{3.84} \times 10^3 \approx 0.39 \times 10^3 \approx 3.9 \times 10^2\).
The Sun is about 390 times as far away as the Moon. For U.S. units, the Moon distance is about \(2.39 \times 10^5\) miles, using 1 mile \(\approx 1{,}609\) m.
Negative exponents work the same way: \(8 \times 10^{-5}\) m (a hair’s width) is ten times larger than \(8 \times 10^{-6}\) m (a red blood cell), because the exponents differ by 1. Careful: with negative exponents, \(10^{-3}\) is larger than \(10^{-5}\), since \(0.001 > 0.00001\).
Key takeaways
- \(a^n\) means \(a\) multiplied by itself \(n\) times; \(a\) is the base and \(n\) the exponent.
- Product rule: \(a^m \times a^n = a^{m+n}\). Quotient rule: \(\dfrac{a^m}{a^n} = a^{m-n}\). Same base only.
- Power of a power: \((a^m)^n = a^{mn}\), and \((ab)^n = a^n b^n\).
- \(a^0 = 1\) and \(a^{-n} = \dfrac{1}{a^n}\) for \(a \neq 0\); a negative exponent flips the base, it does not make the value negative.
- Scientific notation: \(a \times 10^n\) with \(1 \le |a| < 10\); positive \(n\) for big numbers, negative \(n\) for tiny ones.
- To add or subtract, match the powers of 10 first; to multiply or divide, handle the powers of 10 with the exponent rules.
- To compare, look at the exponents first, then at the decimal parts; divide to get “how many times”.
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