1.2 Exponents and Radicals
1.2.1 Learning Objectives
By the end of this section, you should be able to:
- apply the laws of exponents;
- write and interpret numbers in scientific notation;
- simplify radicals and combine like radicals;
- use rational exponents; and
- rationalize denominators.
1.2.2 Integer Exponents and Exponent Laws
For a positive integer \(n\),
\[ a^n=\underbrace{a\cdot a\cdots a}_{n\text{ factors}}. \]
If \(a\ne0\), then \(a^0=1\) and \(a^{-n}=1/a^n\). The main exponent laws are
\[ a^ma^n=a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn}, \]
\[ (ab)^n=a^nb^n,\qquad \left(\frac ab\right)^n=\frac{a^n}{b^n}. \]
A negative exponent indicates a reciprocal, not a negative value.
Parentheses matter: \((-3)^4=81\), but \(-3^4=-(3^4)=-81\).
Example 1.7 Exponent Laws
\[ \frac{(3x^2y^{-1})^2}{9x^{-1}y} =x^5y^{-3} =\boxed{\frac{x^5}{y^3}}. \]
Exercise 1.11 Simplify \(a^6a^{-2}\).
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Answer: \(a^4\)
Exercise 1.12 Rewrite \(x^{-3}/y^{-2}\) using positive exponents.
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Answer: \(y^2/x^3\)
1.2.3 Scientific Notation
A number is in scientific notation when it has the form
\[ a\times10^n,\qquad 1\le |a|<10. \]
Example 1.8 Scientific Notation
\[ 0.0000725=\boxed{7.25\times10^{-5}}. \]
Exercise 1.13 Write \(6.08\times10^7\) in standard form.
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Answer: \(60\,800\,000\)
1.2.4 Radicals and \(n\)th Roots
The expression \(\sqrt[n]{a}\) is the principal \(n\)th root of \(a\). Even roots require nonnegative radicands in the real number system, while odd roots can have negative radicands. For even \(n\),
\[ \sqrt[n]{a^n}=|a|. \]
When the indicated roots exist,
\[ \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}, \qquad \sqrt[n]{\frac ab}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}. \]
Example 1.9 Simplifying Radicals
\[ 3\sqrt{48}-2\sqrt{27} =12\sqrt3-6\sqrt3 =\boxed{6\sqrt3}. \]
Exercise 1.14 Simplify \(\sqrt{72}\).
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Answer: \(6\sqrt2\)
Exercise 1.15 Simplify \(5\sqrt2+\sqrt8\).
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Answer: \(7\sqrt2\)
1.2.5 Rational Exponents
For a rational exponent,
\[ a^{m/n}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m. \]
Example 1.10 Rational Exponents
\[ 64^{2/3}=(\sqrt[3]{64})^2=4^2=\boxed{16}. \]
Exercise 1.16 Evaluate \(81^{3/4}\).
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Answer: \(27\)
1.2.6 Combining Radicals and Rationalizing Denominators
Only like radicals can be combined. A radical denominator can be removed by multiplying by a suitable radical or, for a binomial, its conjugate.
Example 1.11 Rationalizing
\[ \frac4{3-\sqrt5}\cdot\frac{3+\sqrt5}{3+\sqrt5} =\frac{4(3+\sqrt5)}{4} =\boxed{3+\sqrt5}. \]
1.2.7 Conceptual Takeaways
- Exponent laws extend to zero, negative, and rational exponents when expressions are defined.
- Negative exponents represent reciprocals.
- Even roots require nonnegative radicands in the real number system.
- \(\sqrt{a^2}=|a|\).
- Radical notation and rational-exponent notation are equivalent.
- Only like radicals can be combined.
1.2.8 Skills You Should Be Able to Do
- Apply exponent laws and eliminate negative exponents.
- Use scientific notation.
- Simplify \(n\)th roots and combine like radicals.
- Convert between radical and exponential notation.
- Rationalize denominators.
1.2.9 Practice Problems with Solutions
Simplify \(m^5m^{-8}\).
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\(m^5m^{-8}=m^{-3}=\boxed{1/m^3}\).
Simplify \(\frac{(2a^3b^{-2})^2}{4ab^{-1}}\).
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\[ \frac{4a^6b^{-4}}{4ab^{-1}} =a^5b^{-3} =\boxed{\frac{a^5}{b^3}}. \]
Write \(0.000000391\) in scientific notation.
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\(\boxed{3.91\times10^{-7}}\)
Simplify \(\sqrt{180}\).
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\(\sqrt{180}=\sqrt{36\cdot5}=\boxed{6\sqrt5}\)
Simplify \(\sqrt[3]{54x^4}\).
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\[ \sqrt[3]{54x^4} =\sqrt[3]{27x^3(2x)} =\boxed{3x\sqrt[3]{2x}}. \]
Evaluate \(32^{4/5}\).
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\(32^{4/5}=(\sqrt[5]{32})^4=2^4=\boxed{16}\)
Simplify \(4\sqrt{12}+\sqrt{27}-2\sqrt3\).
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\[ 8\sqrt3+3\sqrt3-2\sqrt3=\boxed{9\sqrt3}. \]
Rationalize \(\frac5{2+\sqrt3}\).
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\[ \frac5{2+\sqrt3}\cdot\frac{2-\sqrt3}{2-\sqrt3} =\boxed{10-5\sqrt3}. \]