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Section 9.6 Chapter 9 Summary and Review

Subsection Lesson 9.1 Laws of Exponents

  • First Law of Exponents

    \begin{equation*} \blert{a^m \cdot a^n = a^{m+n}} \end{equation*}

    Second Law of Exponents

    \begin{gather*} \blert{\dfrac{a^m}{a^n} = a^{m-n}~~~~(n \lt m)}\\ \blert{\dfrac{a^m}{a^n} = \dfrac{1}{a^{n-m}}~~~~(n \gt m)} \end{gather*}
  • Third Law of Exponents.

    To raise a power to a power, keep the same base and multiply the exponents. In symbols,

    \begin{equation*} \blert{(a^m)^n = a^{mn}} \end{equation*}
  • Fourth Law of Exponents.

    To raise a product to a power, raise each factor to the power. In symbols,

    \begin{equation*} \blert{(ab)^n = a^nb^n} \end{equation*}
  • Fifth Law of Exponents.

    To raise a quotient to a power, raise both the numerator and the denominator to the power. In symbols,

    \begin{equation*} \blert{(\dfrac{a}{b})^n = \dfrac{a^n}{b^n}} \end{equation*}

Subsection Lesson 9.2 Negative Exponents and Scientific Notation

  • Zero as an Exponent.
    \begin{equation*} \blert{a^0=1,~~~~\text{if}~~~~a \not= 0} \end{equation*}
  • Negative Exponents.
    \begin{equation*} \blert{a^{-n} = \dfrac{1}{a^n}~~~~\text{if}~~~~a \not= 0} \end{equation*}
  • A negative power is the reciprocal of the corresponding positive power. A negative exponent does not mean that the power is negative.
  • The laws of exponents apply to negative exponents. In particular, if we allow negative exponents, we can write the second law as a single rule.

    \begin{equation*} \dfrac{a^m}{a^n} = a^{m-n},~~~~a \not=0 \end{equation*}
  • A number is in scientific notation if it is expressed as the product of a number between 1 and 10 and a power of 10.
  • To write a number in scientific notation, we first position the decimal point and then determine the correct power of 10.
  • We can often obtain a quick estimate for a calculation by converting each figure to scientific notation and rounding to just one or two digits.

Subsection Lesson 9.3 Properties of Radicals

  • Product Rule for Radicals

    \begin{equation*} \text{If}~~a,~b \ge 0,~~~\text{then}~~~~\blert{\sqrt{ab}=\sqrt{a}\sqrt{b}} \end{equation*}

    Quotient Rule for Radicals

    \begin{equation*} \text{If}~~a \ge 0,~b \gt 0~~~\text{then}~~~~\blert{\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}} \end{equation*}
  • It is just as important to remember that we do not have a sum or difference rule for radicals. That is, in general,

    \begin{gather*} \sqrt{a+b} \not= \sqrt{a}+\sqrt{b}\\ \sqrt{a-b} \not= \sqrt{a}-\sqrt{b} \end{gather*}
  • To Simplify a Square Root.
    1. Factor any perfect squares from the radicand.
    2. Use the product rule to write the radical as a product of two square roots.
    3. 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.

Subsection Lesson 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.

Subsection Lesson 9.5 Equations with Radicals

  • A radical equation is one in which the variable appears under a radical.
  • To Solve a Radical Equation.
    1. Isolate the radical on one side of the equation.
    2. Square both sides of the equation.
    3. 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.

Subsection Review Questions

Use complete sentences to answer the questions.

  1. State the third law of exponents, and compare with the first law.
  2. State the fourth and fifth laws of exponents and give examples.
  3. Give the definition of \(a^{-n}\) and give an example.
  4. State the second law of exponents.
  5. Explain how the quotient \(\dfrac{2^5}{2^5}\) illustrates the definition of \(a^0\text{.}\)
  6. Describe the form of a number written in scientific notation.
  7. State two properties of radicals that are useful in simplifying radical expressions.
  8. State two similar "rules" for radicals that are false.
  9. Explain how to simplify a square root to a classmate who was absent that day.
  10. When can you simplify a sum or difference of square roots? How?
  11. Explain how to rationalize the denominator of a fraction.
  12. How do we solve a radical equation?

Subsection Review Problems

Exercises Exercises

For Problems 1–8, simplify the expression.

1.
  1. \(a^4 \cdot a^6\)
  2. \((a^4)^6\)
2.
  1. \(\dfrac{a^4}{a^6}\)
  2. \(\dfrac{a^6}{a^4}\)
3.
  1. \((2a^2)^3\)
  2. \(2a^2(a^2)^3\)
4.
  1. \((\dfrac{-3u}{v^2})^4\)
  2. \(\dfrac{-3u^4}{v^2(v^4)}\)
5.

\(-4x(-2x^2)^3\)

6.

\(-3w^2(-w^3)^2\)

7.

\(4t^2(t^2)^3-(6t^4)^2\)

8.

\((3v)^3(-v^3)-(2v)^2(-v^4)\)

For Problems 9–18, simplify and write without negative exponents.

9.
  1. \(3x^{-2}\)
  2. \((3x)^{-2}\)
10.
  1. \((4y)^0\)
  2. \(4y^0\)
11.
  1. \(\left(\dfrac{5}{z}\right)^{-2}\)
  2. \(\dfrac{5}{z^{-2}}\)
12.
  1. \(\dfrac{16c^{-4}}{8c^{-8}}\)
  2. \(\dfrac{16c^{-4}}{-8c^8}\)
13.

\(3p^{-4}(2p^{-3})\)

14.

\(2q^{-4}(2q)^{-3}\)

15.

\(\dfrac{(4k^{-3})^2}{2k^{-5}}\)

16.

\(\dfrac{6h^{-4}(2h^{-2})}{3h^{-3}}\)

17.

\(5g^{-6}(g^{-3})^{-2}\)

18.

\((8n)^{-2}(n^{-3})^{-4}\)

For Problems 19–24, write in scientific notation.

19.

\(586,000\)

20.

\(12,400,000\)

21.

\(0.0007\)

22.

\(0.000~009\)

23.

\(483 \times 10^3\)

24.

\(0.0035 \times 10^2\)

For Problems 25–28, use scientific notation to compute.

25.

\((48,000,000)(380,000,000)\)

26.

\((0.000~002~41)(1,900,000,000)\)

27.

\(\dfrac{0.000~000~005)}{0.000~2}\)

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?

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?

For Problems 31–38, simplify the radical if possible.

31.
  1. \(\sqrt{4x^6}\)
  2. \(\sqrt{4+x^6}\)
  3. \(\sqrt{(4+x)^6}\)
32.
  1. \(\sqrt{1-w^9}\)
  2. \(\sqrt{1-w^8}\)
  3. \(\sqrt{-w^8}\)
33.

\(-\sqrt{27m^5}\)

34.

\(\pm \sqrt{98q^{99}}\)

35.

\(\sqrt{\dfrac{a^3c}{16}}\)

36.

\(\sqrt{\dfrac{50b^7}{2g^4}}\)

37.

\(\dfrac{2}{3}b\sqrt{12b^3}\)

38.

\(\dfrac{4}{3a^2}\sqrt{45a^3}\)

For Problems 39–46, simplify the expression.

39.

\(3\sqrt{24}+2\sqrt{18}-5\sqrt{6}\)

40.

\(2z\sqrt{x}-3\sqrt{x^3}-6\sqrt{x}\)

41.

\(\dfrac{\sqrt{54w^{12}}}{\sqrt{9w^6}}\)

42.

\(\dfrac{\sqrt{24n^3}}{\sqrt{6n^5}}\)

43.

\(\dfrac{6-3\sqrt{12}}{3}\)

44.

\(\dfrac{\sqrt{8}-\sqrt{12}}{6}\)

45.

\(\dfrac{2}{3}-\dfrac{\sqrt{3}}{2}\)

46.

\(\dfrac{2\sqrt{3}}{5}-1\)

For Problems 47–48, solve by extraction of roots.

47.

\((3a-2)^2=24\)

48.

\(5(2d+1)^2=90\)

For Problems 49–50, solve for the indicated variable.

49.

\(2a^2+4b^2=c^2,~~~~\) for \(b\)

50.

\(25w^2-k=16m,~~~~\) for \(w\)

For Problems 51–56, multiply and simplify.

51.

\(\sqrt{3}(\sqrt{2}-\sqrt{6})\)

52.

\(3\sqrt{2}(8\sqrt{6}-6\sqrt{12})\)

53.

\((2-\sqrt{d})(2+\sqrt{d})\)

54.

\((5-3\sqrt{2})(3+\sqrt{2})\)

55.

\((\sqrt{7}+3)^2\)

56.

\((3\sqrt{t}+1)^2\)

For Problems 51–56, simplify the expression and rationalize the denominator if necessary.

57.

\(\dfrac{2}{\sqrt{x}}\)

58.

\(\sqrt{\dfrac{3a}{b}}\)

59.

\(\dfrac{2\sqrt{5}}{\sqrt{8}}\)

60.

\(\dfrac{a\sqrt{32}}{\sqrt{2a}}\)

61.

\(\dfrac{2}{\sqrt{7}}+\dfrac{3\sqrt{7}}{7}\)

62.

\(\dfrac{1}{2\sqrt{3}}-\dfrac{1}{3\sqrt{2}}\)

For Problems 63–64, verify by substitution that the given value is a solution of the equation.

63.

\(2x^2-2x-3=0,~~x= \dfrac{1+\sqrt{7}}{2}\)

64.

\(x^2+4x-1=0,~~x= 2-\sqrt{5}\)

For Problems 65–66, find the length of the third side of the right triangle.

65.
right triangle
66.
right triangle

For Problems 67–72, solve.

67.

\(3\sqrt{x+2}-4=5\)

68.

\(\sqrt{x-3}+4=2\)

69.

\(\sqrt{2x+1}=x-7\)

70.

\(4\sqrt{4x+1}=5x+2\)

71.

\(\sqrt[3]{3x+2}-4=1\)

72.

\(9-4\sqrt[3]{1-2x}=17\)

73.

The time it takes for a pendulum to complete one full swing, from right to left and back again, is given in seconds by the formula

\begin{equation*} T= 2\pi \sqrt{\dfrac{L}{32}} \end{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?

74.

The velocity, \(v\text{,}\) of a satellite orbitting the earth is given in miles per hour by

\begin{equation*} v=\sqrt{\dfrac{1.24 \times 10^{12}}{R+h}} \end{equation*}

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?