Use the product rule to write the radical as a product of two square roots.
Simplify the square root of the perfect square.
Finding a decimal approximation for a radical is not the same as simplifying the radical
To take the square root of an even power we divide the exponent by 2.
Square roots with identical radicands are called like radicals.
We can add or subtract like radicals in the same way that we add or subtract like terms, namely by adding or subtracting their coefficients.
SubsectionLesson 9.4 Operations on Radicals
We can use the product rule to multiply radicals together: \(\sqrt{a} \sqrt{b} = \sqrt{ab}\)
We can use the quotient rule to simplify quotients of square roots: \(\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}\)
We can multiply numerator and denominator of a fraction by the same root that appears in the denominator to eliminate the radical from the denominator. This process is called rationalizing the denominator.
SubsectionLesson 9.5 Equations with Radicals
A radical equation is one in which the variable appears under a radical.
To Solve a Radical Equation.
Isolate the radical on one side of the equation.
Square both sides of the equation.
Continue as usual to solve for the variable.
The technique of squaring both sides may introduce extraneous solutions.
If a radical equation involves several terms, it is easiest to isolate the radical term on one side of the equation before squaring both sides.
To solve an equation in which the variable appears under a cube root, we isolate the cube root, then cube both sides of the equation.
We do not have to check for extraneous solutions when we cube both sides of an equation.
SubsectionReview Questions
Use complete sentences to answer the questions.
State the third law of exponents, and compare with the first law.
State the fourth and fifth laws of exponents and give examples.
Give the definition of \(a^{-n}\) and give an example.
State the second law of exponents.
Explain how the quotient \(\dfrac{2^5}{2^5}\) illustrates the definition of \(a^0\text{.}\)
Describe the form of a number written in scientific notation.
State two properties of radicals that are useful in simplifying radical expressions.
State two similar "rules" for radicals that are false.
Explain how to simplify a square root to a classmate who was absent that day.
When can you simplify a sum or difference of square roots? How?
Explain how to rationalize the denominator of a fraction.
How do we solve a radical equation?
SubsectionReview Problems
ExercisesExercises
Exercise Group.
For Problems 1–8, simplify the expression.
1.
\(\displaystyle a^4 \cdot a^6\)
\(\displaystyle (a^4)^6\)
Answer.
\(\displaystyle a^{10}\)
\(\displaystyle a^{24}\)
2.
\(\displaystyle \dfrac{a^4}{a^6}\)
\(\displaystyle \dfrac{a^6}{a^4}\)
3.
\(\displaystyle (2a^2)^3\)
\(\displaystyle 2a^2(a^2)^3\)
Answer.
\(\displaystyle 8a^{6}\)
\(\displaystyle 2a^{8}\)
4.
\(\displaystyle (\dfrac{-3u}{v^2})^4\)
\(\displaystyle \dfrac{-3u^4}{v^2(v^4)}\)
5.
\(-4x(-2x^2)^3\)
Answer.
\(32x^7\)
6.
\(-3w^2(-w^3)^2\)
7.
\(4t^2(t^2)^3-(6t^4)^2\)
Answer.
\(-32t^8\)
8.
\((3v)^3(-v^3)-(2v)^2(-v^4)\)
Exercise Group.
For Problems 9–18, simplify and write without negative exponents.
9.
\(\displaystyle 3x^{-2}\)
\(\displaystyle (3x)^{-2}\)
Answer.
\(\displaystyle \dfrac{3}{x^2} \)
\(\displaystyle \dfrac{1}{9x^2} \)
10.
\(\displaystyle (4y)^0\)
\(\displaystyle 4y^0\)
11.
\(\displaystyle \left(\dfrac{5}{z}\right)^{-2}\)
\(\displaystyle \dfrac{5}{z^{-2}}\)
Answer.
\(\displaystyle \dfrac{z^2}{25} \)
\(\displaystyle 5z^2 \)
12.
\(\displaystyle \dfrac{16c^{-4}}{8c^{-8}}\)
\(\displaystyle \dfrac{16c^{-4}}{-8c^8}\)
13.
\(3p^{-4}(2p^{-3})\)
Answer.
\(\dfrac{6}{p^7} \)
14.
\(2q^{-4}(2q)^{-3}\)
15.
\(\dfrac{(4k^{-3})^2}{2k^{-5}}\)
Answer.
\(\dfrac{8}{k} \)
16.
\(\dfrac{6h^{-4}(2h^{-2})}{3h^{-3}}\)
17.
\(5g^{-6}(g^{-3})^{-2}\)
Answer.
5
18.
\((8n)^{-2}(n^{-3})^{-4}\)
Exercise Group.
For Problems 19–24, write in scientific notation.
19.
\(586,000\)
Answer.
\(5.86\times 10^{5} \)
20.
\(12,400,000\)
21.
\(0.0007\)
Answer.
\(7\times 10^{-4} \)
22.
\(0.000~009\)
23.
\(483 \times 10^3\)
Answer.
\(4.83\times 10^{5} \)
24.
\(0.0035 \times 10^2\)
Exercise Group.
For Problems 25–28, use scientific notation to compute.
25.
\((48,000,000)(380,000,000)\)
Answer.
\(18,240,000,000,000,000 \)
26.
\((0.000~002~41)(1,900,000,000)\)
27.
\(\dfrac{0.000~000~005)}{0.000~2}\)
Answer.
\(0.000\,025\)
28.
\(\dfrac{38,500,000}{(0,000~8)(0.001~7)}\)
29.
One atomic unit is equal to \(1.66 \times 10^{-31}\) kilogram. What is the mass of \(6.02 \times 10^{23}\) atomic units?
Answer.
\(9.99\times 10^{-4} \) kg
30.
The mass of an electron is \(9.11 \times 10^{-31}\) kilogram, and the mass of a proton is \(1.67 \times 10^{-27}\) kilogram. How many electrons would you need to match the mass of one proton?
Exercise Group.
For Problems 31–38, simplify the radical if possible.
31.
\(\displaystyle \sqrt{4x^6}\)
\(\displaystyle \sqrt{4+x^6}\)
\(\displaystyle \sqrt{(4+x)^6}\)
Answer.
\(\displaystyle 2x^3 \)
Cannot be simplified
\(\displaystyle (4+x)^3 \)
32.
\(\displaystyle \sqrt{1-w^9}\)
\(\displaystyle \sqrt{1-w^8}\)
\(\displaystyle \sqrt{-w^8}\)
33.
\(-\sqrt{27m^5}\)
Answer.
\(-3m^2\sqrt{3m} \)
34.
\(\pm \sqrt{98q^{99}}\)
35.
\(\sqrt{\dfrac{a^3c}{16}}\)
Answer.
\(\dfrac{a\sqrt{ac}}{4} \)
36.
\(\sqrt{\dfrac{50b^7}{2g^4}}\)
37.
\(\dfrac{2}{3}b\sqrt{12b^3}\)
Answer.
\(\dfrac{4b^2\sqrt{3b}}{3} \)
38.
\(\dfrac{4}{3a^2}\sqrt{45a^3}\)
Exercise Group.
For Problems 39–46, simplify the expression.
39.
\(3\sqrt{24}+2\sqrt{18}-5\sqrt{6}\)
Answer.
\(\sqrt{6}+6\sqrt{2} \)
40.
\(2z\sqrt{x}-3\sqrt{x^3}-6\sqrt{x}\)
41.
\(\dfrac{\sqrt{54w^{12}}}{\sqrt{9w^6}}\)
Answer.
\(w^3\sqrt{6} \)
42.
\(\dfrac{\sqrt{24n^3}}{\sqrt{6n^5}}\)
43.
\(\dfrac{6-3\sqrt{12}}{3}\)
Answer.
\(2-2\sqrt{3} \)
44.
\(\dfrac{\sqrt{8}-\sqrt{12}}{6}\)
45.
\(\dfrac{2}{3}-\dfrac{\sqrt{3}}{2}\)
Answer.
\(\dfrac{4-3\sqrt{3}}{6} \)
46.
\(\dfrac{2\sqrt{3}}{5}-1\)
Exercise Group.
For Problems 47–48, solve by extraction of roots.
47.
\((3a-2)^2=24\)
Answer.
\(\dfrac{2\pm2\sqrt{6}}{3} \)
48.
\(5(2d+1)^2=90\)
Exercise Group.
For Problems 49–50, solve for the indicated variable.
49.
\(2a^2+4b^2=c^2,~~~~\) for \(b\)
Answer.
\(\dfrac{\pm\sqrt{c^2-2a^2}}{2} \)
50.
\(25w^2-k=16m,~~~~\) for \(w\)
Exercise Group.
For Problems 51–56, multiply and simplify.
51.
\(\sqrt{3}(\sqrt{2}-\sqrt{6})\)
Answer.
\(\sqrt{6}-3\sqrt{2} \)
52.
\(3\sqrt{2}(8\sqrt{6}-6\sqrt{12})\)
53.
\((2-\sqrt{d})(2+\sqrt{d})\)
Answer.
\(4-d \)
54.
\((5-3\sqrt{2})(3+\sqrt{2})\)
55.
\((\sqrt{7}+3)^2\)
Answer.
\(16+6\sqrt{7}\)
56.
\((3\sqrt{t}+1)^2\)
Exercise Group.
For Problems 51–56, simplify the expression and rationalize the denominator if necessary.
57.
\(\dfrac{2}{\sqrt{x}}\)
Answer.
\(\dfrac{2\sqrt{x}}{x} \)
58.
\(\sqrt{\dfrac{3a}{b}}\)
59.
\(\dfrac{2\sqrt{5}}{\sqrt{8}}\)
Answer.
\(\dfrac{\sqrt{10}}{2} \)
60.
\(\dfrac{a\sqrt{32}}{\sqrt{2a}}\)
61.
\(\dfrac{2}{\sqrt{7}}+\dfrac{3\sqrt{7}}{7}\)
Answer.
\(\dfrac{5\sqrt{7}}{7} \)
62.
\(\dfrac{1}{2\sqrt{3}}-\dfrac{1}{3\sqrt{2}}\)
Exercise Group.
For Problems 63–64, verify by substitution that the given value is a solution of the equation.
where \(L\) is the length of the pendulum in feet. The longest pendulum in the world is a reconstruction of Foucault’s pendulum in the Convention Center in Portland, Oregon. The pendulum weighs 900 pounds and takes 10.54 seconds to complete one full swing. To the nearest foot, how long is the pendulum?
Answer.
90 ft
74.
The velocity, \(v\text{,}\) of a satellite orbitting the earth is given in miles per hour by
where \(h\) is the altitude of the satellite in miles, and \(R\) is the radius of the earth, about 3960 miles. The Russian space station Mir has an orbital velocity of 17,187 miles per hour. What is its altitude?
