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CS221 Exponents

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Revision

You will need to know these rules and notation:

  1. a^b \cdot a^c=a^{b+c}
  2. \sqrt[n]{a} = a^{\frac{1}{n}}
  3. (ab)^n =a^nb^n
  4. (a^b)^c=a^{bc}
  5. a^{-n} = \frac{1}{a^n}
  6. \frac{a^b}{a^c} = a^{b-c}

and the situations they are used in, as shown in the following sections.

Rule: a^b \cdot a^c=a^{b+c}

Examples

  1. a\cdot a^n=a^1a^{n}=a^{n+1}
  2. 2\cdot2^5 = 2^6
  3. 2\cdot2^n = 2^{n+1}
  4. 2\cdot2^{n-1} = 2^n
  5. 4\cdot2^n = 2^2\cdot2^{n}=2^{n+2}
  6. 2^n + 2\cdot2^n + 2^n = 1\cdot2^n +2\cdot2^n + 1\cdot2^n = (1+2+1)\cdot2^n = 4\cdot2^n = 2^{n+2}
  7. (-1)\cdot(-1)^n = (-1)^{n+1}
  8. -(-1)^n =(-1)(-1)^n=(-1)^{n+1} - be careful with those parentheses as without, -1^n = -1 for all integers n, whereas (-1)^n is -1 for odd n and 1 for even n

Exercises

A. Fill in the values in the boxes in the following exercises.

  1. 2^5 = ___
  2. 2^2\cdot2^3 =\square \cdot \square = ___
  3. 2^2\cdot2^3 = 2^{\square}
  4. 4=2^{\square}
  5. 4\cdot2^3 = ___
  6. 4\cdot2^3 =2^\square
  7. -1^3 = ___
  8. -1^{10} = ___
  9. (-1)^3 = ___
  10. (-1)^{10} = ___

B. Simplify the following expressions to a single power.

  1. 3^n + 3^n + 3^n = 3^\square
  2. 6\cdot3^n - 2\cdot3^n - 3^n = 3^\square
  3. 3^{n+1} + 3\cdot3^n + 3^{n+1} = 3^\square
  4. (-1)^n +(-1)^{n+1} +(-1)^n = (-1)^\square
  5. 2^{n+1} + 2\cdot2^{n+1} + 2^{n+1} = 2^\square

Answers

A.

  1. 2^5 = 32
  2. 2^2\cdot2^3 = 4\cdot8=32
  3. 2^2\cdot2^3 = 2^{5}
  4. 4=2^{2}
  5. 4\cdot2^3 =4\cdot8 = 32
  6. 4\cdot2^3 =2^2\cdot2^3=2^5
  7. -1^3 =-(1^3)=-1
  8. -1^{10} = -(1^{10}) = -1
  9. (-1)^3 = -1
  10. (-1)^{10} = 1

B.

  1. 3^n + 3^n + 3^n = 3\cdot3^n = 3^{n+1}
  2. 6\cdot3^n - 2\cdot3^n - 3^n = (6-2-1)\cdot3^n =3\cdot3^n=3^{n+1}
  3. 3^{n+1} + 3\cdot3^n + 3^{n+1} = 3\cdot3^n+3\cdot3^n + 3\cdot3^n = (3+3+3)3^n = 9\cdot3^n = 3^2\cdot3^n = 3^{n+2} or 3^{n+1} + 3\cdot3^n + 3^{n+1} = 3^{n+1} + 3^{n+1} + 3^{n+1} = 3\cdot3^{n+1} = 3^{1+n+1} = 3^{n+2}
  4. (-1)^n +(-1)^{n+1} +(-1)^n = (-1)^n + (-1)(-1)^n + (-1)^n = (-1)^n -(-1)^n + (-1)^n = (-1)^n . Note that by adding (or subtracting) any multiple of 2 to the power will result in the same value so (-1)^{n+2} would also be correct as would (-1)^{n+276}, although it wouldn't be particularly good to write it in such a form.
  5. 2^{n+1} + 2\cdot2^{n+1} + 2^{n+1} = (1+2+1)\cdot2^{n+1}= 4\cdot2^{n+1}=2^2\cdot2^{n+1}=2^{n+3}

Notation: \sqrt[n]{a} = a^{\frac{1}{n}}

Examples

  1. \sqrt{25} = 25^{\frac{1}{2}}= 5
  2. 36^{\frac{1}{2}}=6
  3. \sqrt[3]{125} = 125^{\frac{1}{3}}= 5
  4. 7^{\frac{1}{2}} = \sqrt{7}

Exercises

  1. 64^{\frac{1}{2}}= ___
  2. 8^{\frac{1}{3}}= ___
  3. \sqrt{5} =5^\square

Answers

  1. 64^{\frac{1}{2}}= \sqrt{64}=8
  2. 8^{\frac{1}{3}}=\sqrt[3]8=2
  3. \sqrt{5} =5^{\frac{1}{2}}

Rule: (ab)^n =a^nb^n

Examples

  1. 6^n = (2\cdot3)^n=2^n\cdot3^n
  2. (2x)^3 = 2^3x^3 = 8x^3
  3. (-2)^n = (-1\cdot 2)^n = (-1)^n\cdot2^n

Exercises

  1. (3x)^2 =\square x^\square
  2. (-x)^n = (\square)^n\square^n

Answers

  1. (3x)^2 =3^2x^2 = 9x^2
  2. (-x)^n = (-1)^nx^n

Rule: (a^b)^c=a^{bc}

Examples

  1. Applying the rule, (2^3)^2 = 2^{2\cdot3} = 2^6 = 64, which we can check is correct by calculating 2^3 =8 and then squaring, which gives (2^3)^2=8^2=64, as before.
  2. (\sqrt{x})^k = (x^{\frac{1}{2}})^k = x^{k\cdot\frac{1}{2}} =x^{\frac{k}{2}}

Exercises

  1. (2^2)^3 = ( ___ )^3 = ___
  2. (2^2)^3 = 2^\square = ___
  3. (x^4)^3 = x^\square
  4. (\sqrt{5})^k = 5^\square
  5. (\sqrt{5})^{k-1} = 5^\square

Answers

  1. (2^2)^3 = 4^3 = 64
  2. (2^2)^3 = 2^{2\cdot3}=2^6 = 64
  3. (x^4)^3 = x^{4\cdot3}=x^{12}
  4. (\sqrt{5})^k =(5^{\frac{1}{2}})^{k}=5^{\frac{1}{2}k}= 5^{\frac{k}{2}}
  5. (\sqrt{5})^{k-1} =(5^{\frac{1}{2}})^{k-1}=5^{\frac{1}{2}(k-1)}=5^{\frac{k-1}{2}}

Notation: a^{-n} = \frac{1}{a^n}

Examples

  1. \frac{1}{5^2} = 5^{-2}
  2. 3^{-2} = \frac{1}{3^2}= \frac{1}{9}
  3. \frac{2}{x} = 2x^{-1}
  4. 3x^{-5}=\frac{3}{x^5}

Exercises

  1. 5^{-1} = ___
  2. 2^{-3} = ___
  3. 3^{-1} = ___
  4. \frac{1}{\sqrt5} = 5^\square

Answers

  1. 5^{-1} = \frac{1}{5}
  2. 2^{-3} = \frac{1}{2^3}=\frac{1}{8}
  3. 3^{-1} = \frac{1}{3}
  4. \frac{1}{\sqrt5} = 5^{-\frac{1}{2}}

Rule: \frac{a^b}{a^c} = a^{b-c}

Examples

  1. Using the rule, \frac{2^6}{2^4} = 2^{6-4} = 2^2 = 4 , which we can check by calculating the values of the numerator and denominator: \frac{2^6}{2^4}= \frac{64}{16} = 4 .
  2. \frac{(\sqrt x)^5}{(\sqrt x)^3} =(\sqrt x)^{5-3} = (\sqrt x)^2 = x

Exercises

  1. \frac{2^4}{2}= \frac{\square}{\square} = ___
  2. \frac{2^4}{2}= 2^\square
  3. \frac{x^k}{x^3} = x^\square
  4. \frac{x^k}{x} = x^\square
  5. \frac{(\sqrt 5)^k}{\sqrt 5} =(\sqrt 5)^{\square-\square} = 5^\square

Answers

  1. \frac{2^4}{2}= \frac{16}{2} = 8
  2. \frac{2^4}{2}= \frac{2^4}{2^1}=2^{4-1}= 2^3
  3. \frac{x^k}{x^3} = x^{k-3}
  4. \frac{x^k}{x} = x^{k-1}
  5. \frac{(\sqrt 5)^k}{\sqrt 5} =(\sqrt 5)^{k-1} =(5^{\frac{1}{2}})^{k-1}=5^{\frac{1}{2}(k-1)}= 5^{\frac{k-1}{2}}