Section 6.5 Radical Expressions and Equations Chapter Review
ΒΆSubsection 6.5.1 Square and th Root Properties
permalinkIn Section 6.1 we defined the square root and th root radicals. When is positive, the expression means a positive number where The square root is just the case where
permalinkWhen is negative, might not be defined. It depends on whether or not is an even number. When is negative and is odd, is a negative number where
permalinkThere are two helpful rules for simplifying radicals.
If and are positive real numbers, and is a positive integer , then we have the following rules:
- Root of a Product Rule
- Root of a Quotient Rule
as long as
Checkpoint 6.5.2.
Subsection 6.5.2 Rationalizing the Denominator
permalinkIn Section 6.2 we covered how to rationalize the denominator when it contains a single square root or a binomial with a square root term.
Example 6.5.3.
Rationalize the denominator of the expressions.
- \begin{align*} \frac{12}{\sqrt{3}}\amp=\frac{12}{\sqrt{3}}\multiplyright{\frac{\sqrt{3}}{\sqrt{3}}}\\ \amp=\frac{12\sqrt{3}}{3}\\ \amp=4\sqrt{3} \end{align*}
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First we will simplify \(\sqrt{75}\text{.}\)
\begin{align*} \frac{\sqrt{5}}{\sqrt{75}}\amp=\frac{\sqrt{5}}{\sqrt{25\cdot 3}}\\ \amp=\frac{\sqrt{5}}{\sqrt{25}\cdot\sqrt{3}}\\ \amp=\frac{\sqrt{5}}{5\sqrt{3}}\\ \end{align*}Now we can rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{3}\text{.}\)
\begin{align*} \amp=\frac{\sqrt{5}}{5\sqrt{3}}\multiplyright{\frac{\sqrt{3}}{\sqrt{3}}}\\ \amp=\frac{\sqrt{15}}{5\cdot 3}\\ \amp=\frac{\sqrt{15}}{15} \end{align*}
Example 6.5.4. Rationalize Denominator Using the Difference of Squares Formula.
Rationalize the denominator in
To remove radicals in \(\sqrt{3}+\sqrt{2}\) with the difference of squares formula, we multiply it with \(\sqrt{3}-\sqrt{2}\text{.}\)
Subsection 6.5.3 Radical Expressions and Rational Exponents
permalinkIn Section 6.3 we learned the rational exponent rule and added it to our list of exponent rules.
Example 6.5.5. Radical Expressions and Rational Exponents.
Simplify the expressions using Fact 6.3.2 or Fact 6.3.9.
\(\begin{aligned}[t] 100^{\sfrac{1}{2}}\amp=\left(\sqrt{100}\right)\\ \amp=10 \end{aligned}\)
\(\begin{aligned}[t] (-64)^{-\sfrac{1}{3}}\amp=\frac{1}{(-64)^{\sfrac{1}{3}}}\\ \amp=\frac{1}{\left(\sqrt[3]{(-64)}\right)}\\ \amp=\frac{1}{-4} \end{aligned}\)
\(\begin{aligned}[t] -81^{\sfrac{3}{4}}\amp=-\left(\sqrt[4]{81}\right)^3\\ \amp=-3^3\\ \amp=-27 \end{aligned}\)
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In this problem the negative can be associated with either the numerator or the denominator, but not both. We choose the numerator.
\begin{align*} \left(-\frac{1}{27}\right)^{\sfrac{2}{3}}\amp=\left(\sqrt[3]{-\frac{1}{27}}\right)^2\\ \amp=\left(\frac{\sqrt[3]{-1}}{\sqrt[3]{27}}\right)^2\\ \amp=\left(\frac{-1}{3}\right)^2\\ \amp=\frac{(-1)^2}{(3)^2}\\ \amp=\frac{1}{9} \end{align*}
Example 6.5.6. More Expressions with Rational Exponents.
Use exponent properties in List 6.3.14 to simplify the expressions, and write all final versions using radicals.
\(\begin{aligned}[t] 7z^{\sfrac{5}{9}}\amp=7\sqrt[9]{z^5} \end{aligned}\)
\(\begin{aligned}[t] \frac{5}{4}x^{-\sfrac{2}{3}}\amp=\frac{5}{4}\cdot\frac{1}{x^{\sfrac{2}{3}}}\\ \amp=\frac{5}{4}\cdot\frac{1}{\sqrt[3]{x^2}}\\ \amp=\frac{5}{4\sqrt[3]{x^2}} \end{aligned}\)
\(\begin{aligned}[t] \left(-9q^5\right)^{\sfrac{4}{5}}\amp=\left(-9\right)^{\sfrac{4}{5}}\cdot\left(q^5\right)^{\sfrac{4}{5}}\\ \amp=\left(-9\right)^{\sfrac{4}{5}}\cdot q^{5\cdot\sfrac{4}{5}}\\ \amp=\left(\sqrt[5]{-9}\right)^4\cdot q^{4}\\ \amp=\left(q\sqrt[5]{-9}\right)^4 \end{aligned}\)
\(\begin{aligned}[t] \sqrt{y^5}\cdot\sqrt[4]{y^2}\amp=y^{\sfrac{5}{2}}\cdot y^{\sfrac{2}{4}}\\ \amp=y^{\sfrac{5}{2}+\sfrac{2}{4}}\\ \amp=y^{\sfrac{10}{4}+\sfrac{2}{4}}\\ \amp=y^{\sfrac{12}{4}}\\ \amp=y^3 \end{aligned}\)
\(\begin{aligned}[t] \frac{\sqrt{t^3}}{\sqrt[3]{t^2}}\amp=\frac{t^{\sfrac{3}{2}}}{t^{\sfrac{2}{3}}}\\ \amp=t^{\sfrac{3}{2}-\sfrac{2}{3}}\\ \amp=t^{\sfrac{9}{6}-\sfrac{4}{6}}\\ \amp=t^{\sfrac{5}{6}}\\ \amp=\sqrt[6]{t^5} \end{aligned}\)
\(\begin{aligned}[t] \sqrt{\sqrt[3]{x}}\amp=\sqrt{x^{\sfrac{1}{3}}}\\ \amp=\left(x^{\sfrac{1}{3}}\right)^{\sfrac{1}{2}}\\ \amp=x^{\sfrac{1}{3}\cdot\sfrac{1}{2}}\\ \amp=x^{\sfrac{1}{6}}\\ \amp=\sqrt[6]{x} \end{aligned}\)
\(\begin{aligned}[t] 5\left(4+a^{\sfrac{1}{2}}\right)^2\amp=5\left(4+a^{\sfrac{1}{2}}\right)\left(4+a^{\sfrac{1}{2}}\right)\\ \amp=5\left(4^2+2\cdot4\cdot a^{\sfrac{1}{2}}+\left(a^{\sfrac{1}{2}}\right)^2\right)\\ \amp=5\left(16+8a^{\sfrac{1}{2}}+a^{\sfrac{1}{2}\cdot 2}\right)\\ \amp=5\left(16+8a^{\sfrac{1}{2}}+a\right)\\ \amp=5\left(16+8\sqrt{a}+a\right)\\ \amp=80+40\sqrt{a}+5a \end{aligned}\)
\(\begin{aligned}[t] -6\left(2p^{-\sfrac{5}{2}}\right)^{\sfrac{3}{5}}\amp=-6\cdot2^{\sfrac{3}{5}}\cdot p^{-\sfrac{5}{2}\cdot\sfrac{3}{5}}\\ \amp=-6\cdot2^{\sfrac{3}{5}}\cdot p^{-\sfrac{3}{2}}\\ \amp=-\frac{6\cdot 2^{\sfrac{3}{5}}}{p^{\sfrac{3}{2}}}\\ \amp=-\frac{6\sqrt[5]{2^3}}{\sqrt{p^3}}\\ \amp=-\frac{6\sqrt[5]{8}}{\sqrt{p^3}} \end{aligned}\)
Subsection 6.5.4 Solving Radical Equations
permalinkIn Section 6.4 we covered solving equations that contain a radical. We learned about extraneous solutions and the need to check our solutions.
Example 6.5.7. Solving Radical Equations.
Solve for in
We will isolate the radical first, and then square both sides.
Because we squared both sides of an equation, we must check both solutions.
It turns out \(6\) is an extraneous solution and \(13\) is a valid solution. So the equation has one solution: \(13\text{.}\) The solution set is \(\{13\}\text{.}\)
Example 6.5.8. Solving Radical Equations that Require Squaring Twice.
Solve the equation for
We cannot isolate two radicals, so we will simply square both sides, and later try to isolate the remaining radical.
Because we squared both sides of an equation, we must check the solution by substituting \(\substitute{16}\) into \(\sqrt{t+9}=-1-\sqrt{t}\text{,}\) and we have:
Our solution did not check so there is no solution to this equation. The solution set is the empty set, which can be denoted \(\{\text{ }\}\) or \(\emptyset\text{.}\)
Exercises 6.5.5 Exercises
Square Root and th Root
5.
Simplify the radical expression or state that it is not a real number
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6.
Simplify the radical expression or state that it is not a real number
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7.
Simplify the radical expression or state that it is not a real number
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8.
Simplify the radical expression or state that it is not a real number
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Rationalizing the Denominator
29.
Rationalize the denominator and simplify the expression.
30.
Rationalize the denominator and simplify the expression.
31.
Rationalize the denominator and simplify the expression.
32.
Rationalize the denominator and simplify the expression.
33.
Rationalize the denominator and simplify the expression.
34.
Rationalize the denominator and simplify the expression.
35.
Rationalize the denominator and simplify the expression.
36.
Rationalize the denominator and simplify the expression.
Radical Expressions and Rational Exponents
37.
Without using a calculator, evaluate the expression.
38.
Without using a calculator, evaluate the expression.
39.
Without using a calculator, evaluate the expression.
40.
Without using a calculator, evaluate the expression.
41.
Without using a calculator, evaluate the expression.
42.
Without using a calculator, evaluate the expression.
43.
Without using a calculator, evaluate the expression.
44.
Without using a calculator, evaluate the expression.
49.
Convert the expression to radical notation.
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50.
Convert the expression to radical notation.
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51.
Convert the expression to radical notation.
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52.
Convert the expression to radical notation.
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53.
Convert the expression to radical notation.
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54.
Convert the expression to radical notation.
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55.
Simplify the expression, answering with rational exponents and not radicals.
56.
Simplify the expression, answering with rational exponents and not radicals.
57.
Simplify the expression, answering with rational exponents and not radicals.
58.
Simplify the expression, answering with rational exponents and not radicals.
59.
Simplify the expression, answering with rational exponents and not radicals.
60.
Simplify the expression, answering with rational exponents and not radicals.
61.
Simplify the expression, answering with rational exponents and not radicals.
62.
Simplify the expression, answering with rational exponents and not radicals.
Solving Radical Equations
63.
Solve the equation.
64.
Solve the equation.
65.
Solve the equation.
66.
Solve the equation.
67.
Solve the equation.
68.
Solve the equation.
69.
Solve the equation.
70.
Solve the equation.
71.
Solve the equation.
72.
Solve the equation.
73.
According to the Pythagorean Theorem, the length of the hypothenuse of a rectangular triangle can be found through the following equation.
If a rectangular triangle has a hypothenuse of and one leg is long, how long is the third side of the triangle?
The third side of the triangle is long.
74.
According to the Pythagorean Theorem, the length of the hypothenuse of a rectangular triangle can be found through the following equation.
If a rectangular triangle has a hypothenuse of and one leg is long, how long is the third side of the triangle?
The third side of the triangle is long.
75.
A pendulum has the length ft. The time period that it takes to once swing back and forth is s. Use the following formula to find its length.
The pendulum is long.
76.
A pendulum has the length ft. The time period that it takes to once swing back and forth is s. Use the following formula to find its length.
The pendulum is long.