Learn – Algebra foundations – Power rules

1 Multiplication and division

1.1 Exercises

1.1.1 Problem set

Simplify the following.

  1. -2x^2x^3

  2. 3x^2y^2x^3y^3

  3. \frac{a^{35}}{a^7}

  4. \frac{9x^{33}y^{12}}{3x^{11}y^{4}}

  5. \frac{3p^{21}}{q^{5}}\times\frac{q^{25}}{9p^{7}}

  6. \frac{5a^{36}}{b^{12}}\times\frac{b^{24}}{25a^6}

  7. \frac{2x^{20}}{y^{14}}\times\frac{y^{21}}{8x^5}

  8. \frac{7p^{14}}{q^{5}}\times\frac{q^{10}}{14p^7}

1.1.2 Problem set

Explain, with a reason, whether each of the following is true or false.

  1. x^{11} \times x^{10} = x^{110}

  2. \frac{p^{21}}{p^{7}} = p^{3}

  3. x^{5} \times y^{7} = xy^{12}

  4. a^{9} \times b^{4} = ab^{13}

  5. pq^{9} = p^{9}q^{9}

1.1.3 Problem set

Explain, with a reason, whether each of the following is true or false.

  1. x^{15} + x^{6} = x^{21}

  2. a^{19} - a^{13} = a^{6}

  3. p^{7} + p^{7} = p^{14}

  4. x^{5} + x^{5} = x^{10}

  5. a^{11} + a^{11} = 2a^{11}

  6. 7p^{5} - 4p^{5} = 3

  7. 5x^{9} - x^{9} = 5

2 Distributive property

2.1 Exercises

2.1.1 Problem set

Simplify the following.

  1. \left(pqr\right)^5

  2. \left(3mn\right)^3

  3. \left(\frac{2}{x}\right)^3

  4. 2(2x)^3

  5. 3(5x)^2

  6. \left(\frac{x}{zy}\right)^{51}

  7. \left(\frac{3a}{4bc}\right)^{17}

2.1.2 Problem set

Simplify the following.

  1. 2\left(\frac{2x}{y}\right)^2

  2. 4\left(\frac{2pq}{3r}\right)^3

  3. 5\left(\frac{7x}{5yz}\right)^2

  4. 3\left(\frac{2mn^{100}}{n^{99}m^2}\right)^3

  5. 5\left(\frac{q^4p}{5q^3}\right)^2

2.1.3 Problem set

Simplify the following.

  1. 2\left(\frac{-3x}{4y}\right)^2

  2. -5\left(\frac{-2pq}{3r}\right)^2

2.1.4 Problem set

Explain, with a reason, whether each of the following is true or false.

  1. (ab)^{21} = a^{21}b^{21}

  2. \frac{p^{17}}{q^{11}} = \left(\frac{p}{q}\right)^6

  3. (x+y)^{15} = x^{15}+y^{15}

  4. (p-q)^{11} = p^{11}-q^{11}

2.1.5 Problem set

Explain, with a reason, whether each of the following is true or false.

  1. \left(\frac{3a}{2b}\right)^{9} = \frac{3a^9}{2b^9}

  2. \left(\frac{5x}{yz}\right)^{7} = \frac{5x^7}{y^7z^7}

  3. \left(\frac{-3a}{4bc}\right)^{2} = \frac{-3^2a^2}{4^2b^2c^2}

  4. \left(\frac{5x}{-3yz}\right)^{4} = \frac{5^4x^4}{-3^4y^4z^4}

  5. (3p-4q)^{5} = 3^5p^5-4^5q^5

  6. (2a+5b)^{7} = 2^7a^7+5^7b^7

2.1.6 Problem set

Explain, with a reason, whether each of the following is true or false.

  1. a^{11}b^{11} = (ab)^{11}

  2. x^{7}y^{7} = (xy)^{14}

  3. p^{9}+q^{9} = (p+q)^{9}

  4. a^{7}-b^{7} = (a-b)^{7}

3 Power of a power

3.1 Exercises

3.1.1 Problem set

Explain, with a reason, whether each of the following is true or false.

  1. (a^5)^2 = a^{10}

  2. (x^7)^2 = x^{49}

  3. p^{8^2} = p^{64}

  4. a^{3^3} = (a^3)^3

  5. x^{2^3} = (x^2)^3

4 Zero and negative exponents

4.1 Exercises

4.1.1 Problem set

Evaluate the following.

  1. 0.1^4

  2. (-0.1)^4

  3. -0.1^4

  4. 10^{-2}

  5. -10^{-2}

  6. (-10)^{-2}

  7. 2^{-4}

  8. -2^{-4}

  9. (-2)^{-4}

4.1.2 Problem set

Evaluate the following.

  1. 5^{-2}

  2. (-5)^{-2}

  3. -5^{-2}

  4. 0.5^{-2}

  5. (-0.5)^{-2}

  6. -0.5^{-2}

4.1.3 Problem set

Evaluate the following.

  1. (\frac{1}{2})^{2}

  2. (\frac{1}{2})^{-2}

  3. (\frac{2}{3})^{3}

  4. (\frac{2}{3})^{-3}

4.1.4 Problem set

Explain, with a reason, whether each of the following is true or false.

  1. 3^0 = 0

  2. 5^0 = 7^0

  3. 9^0 = (-9)^0

  4. a^{-7} = -a^7

  5. x^{-81} = \frac{1}{-x^{81}}

  6. 3p^{-5} = \frac{1}{3p^{5}}

  7. (-3ab)^{-2} = \frac{1}{-9a^2b^2}

4.1.5 Problem set

Simplify the following.

  1. \frac{x^{-5}}{x^{-8}}

  2. \frac{a^{-20}}{a^{-10}}

  3. \frac{p^{-18}}{p^{-13}}

4.1.6 Problem set

Simplify the following.

  1. \frac{x^{-4}}{y^{-7}}

  2. \frac{a^{-5}}{b^{-9}}

  3. \frac{p^{-25}}{q^{-15}}

5 Fractional exponents

5.1 Exercises

5.1.1 Problem set

Evaluate the following.

  1. 9^{1/2}

  2. 27^{1/3}

  3. 16^{1/4}

  4. 0.01^{1/2}

  5. (-8)^{1/3}

  6. (-1000)^{1/3}

  7. (-32)^{1/5}

5.1.2 Problem set

Evaluate the following.

  1. 9^{-1/2}

  2. (-8)^{-1/3}

  3. (-125)^{-1/3}

  4. (-0.001)^{-1/3}

  5. 16^{3/2}

5.1.3 Problem set

Evaluate the following.

  1. 25^{3/2}

  2. 8^{4/3}

  3. 27^{2/3}

  4. \left(\frac{1}{16}\right)^{3/2}

  5. \left(\frac{16}{81}\right)^{3/4}

  6. \left(\frac{125}{8}\right)^{2/3}

5.1.4 Problem set

Evaluate the following.

  1. 4^{0.5}

  2. 4^{-0.5}

  3. 10000^{0.25}

  4. 10000^{-0.25}

  5. 32^{0.2/0.25}

6 Combinations

6.1 Exercises

6.1.1 Problem set

Simplify the following.

  1. \left(\frac{2a}{3b}\right)^{100}\times \left(\frac{3b}{a}\right)^{51}

  2. \left(\frac{5x}{7y}\right)^{40}\times \left(\frac{7y}{5x}\right)^{73}

  3. 4(-5a^{-3}b^3)^2

  4. -5(-2p^{-2}q^2)^4

  5. -2(-3a^{-2}b^2c^{-6})^2

6.1.2 Problem set

Simplify the following.

  1. 3\left(\frac{-2a^{-3}b^2}{a^3}\right)^2

  2. -4\left(\frac{-3a^3b^2}{b^{-3}a^5}\right)^{-2}

  3. -3\left(\frac{-2a^3b^2}{b^{-3}a^5}\right)^{-3}

  4. -2\left(\frac{-3x^3y^2x^4}{y^{-3}x^5y^7}\right)^{-4}

  5. \left(\frac{-2a^2}{b^2}\right)^2\left(\frac{b}{2a}\right)^3

  6. \left(\frac{-x^2}{2y^3z}\right)^2\left(-\frac{x}{3y^2z}\right)^{-2}

7 Solving equations

7.1 Variable in the exponent

7.1.1 Exercises

7.1.1.1 Problem set

Solve for x.

  1. 32 = 2^x

  2. 3^x = 81

  3. 0.0001 = 0.1^x

  4. 16 = (-4)^x

  5. 81 = (-3)^x

  6. 0.01 = (-0.1)^x

  7. -0.001 = (-0.1)^x

  8. -25 = -5^x
7.1.1.2 Problem set

Solve for x.

  1. 3^x = 9^5

  2. 5^x = 25^{11}

  3. 10^x = 1000^{5}

  4. 2^x = 16^4

  5. 0.1^x = 0.001^7

  6. 10^x = 0.1^4

  7. 2^x = 0.5^9

  8. 10^x = 0.001^6
7.1.1.3 Problem set

Solve for x.

  1. 3^{2-x} = 81^2

  2. 2^{-x-7} = 16^{2-x}

  3. 10^{3x+7} = 10000^{x-5}

  4. 5^{2x+9} = 125^{8-x}

  5. 0.1^{11-x} = 10^{2x-5}

  6. 2^{3x} = 0.5^{8-x}

  7. 5^{x+4} = 0.2^{x+10}
7.1.1.4 Problem set

Solve for x.

  1. 8^x = 4^9

  2. 25^x = 125^4

  3. 4^{3+x} = 8^{x-10}

  4. 9^{2x-1} = 27^{4-x}

  5. 125^{-3-5x} = 25^{8-10x}

  6. 100^{7-5x} = 0.1^{3x+7}

  7. 8^{4x+3} = 0.5^{13-x}
7.1.1.5 Problem set

Solve for x.

  1. 9^{2x}\times 9^5 = 3^{x}\times 3^{19}

  2. (0.1)^{5-x}\times (0.01)^{2x-10}=(0.001)^4

  3. \frac{3^{x-1}}{9^{1-x}} = 27^{50}

  4. 5^{2x+4} = \frac{125^{3-x}}{25^{10}}

  5. 2^{x^2} = 8^3
7.1.1.6 Problem set

Solve for x.

  1. 5^{2x-3} = 0.2^{2x-5}

  2. \frac{2^{x+1}}{0.5^{2x+1}} = 1

  3. (a^{x})^{6x} = (a^3)^8
7.1.1.7 Problem set

Solve for x.

  1. 5\times 2^{101}+12\times 2^{99} = 2^x

  2. 20\times 4^{50}+24\times 2^{99} = 8^x

  3. 3^{27^x} = 27^{3^x}

  4. 2^{256^x} = 256^{2^x}

  5. 7^{\left(7^7\right)^x} = \left(7^7\right)^{7^x}

7.2 Variable in the base

7.2.1 Exercises

7.2.1.1 Problem set

Solve for x.

  1. 16 = x^4

  2. -125 = x^3

  3. 0.008 = x^3

  4. 0.0001 = x^2

  5. 0.5 = x^{-1}

  6. -0.2 = x^{-1}

  7. -125 = -x^3
7.2.1.2 Problem set

Solve for x.

  1. 81^3 = x^6

  2. 2^{100}\times 3^{50} = x^{25}

  3. 5^{22}\times\left(x^2\right)^5=\frac{\left(x^3\right)^7}{7^{11}}

  4. x^5\times 5^{15} = x^{20}

  5. x^{50} = x^{-50}

  6. \frac{x^{39}}{x^3} = x^{13}
7.2.1.3 Problem set

Solve for x.

  1. 5x^3 = 2x^3 + 24

  2. 5x^3 = -81 + 2x^3

8 Miscellaneous

8.1 Exercises

8.1.1 Problem set

  1. Name the smallest and biggest of the following: (0.99)^{100}, (0.99)^{200}, (0.99)^{300}

  2. Name the smallest and biggest of the following: (0.1)^{100}, (0.01)^{40}, (0.001)^{30}

  3. \left(\left(\left((-1)^{-1}\right)^{-1}\right)^{-1}\right)^{-1}

  4. \left(\left(\left(\left((-0.1)^{-1}\right)^{-1}\right)^{-1}\right)^{-1}\right)^{-1}