Exponents

Integer exponents, rules of exponents, scientific notation and exponential equations. 4 problems with step-by-step solutions.

Solve each problem in your notebook first, then open the solution.

Problem 1 Integer exponents

a) Evaluate \((-2)^3 + 3^{-2}\cdot 9 + 5^0 - (-1)^{11}\).

b) Evaluate \(0.5^{-2} + 2^{-1} + (-3)^{-2}\).

c) Write without negative exponents: \(\dfrac{a^{-2}b^3}{c^{-1}}\), \((x^{-1} + y^{-1})^{-1}\).

Show solution

a) \(-8 + 1 + 1 + 1 = -5\).

b) \(4 + \frac12 + \frac19 = \frac{72 + 9 + 2}{18} = \frac{83}{18}\).

c) \(\frac{b^3 c}{a^2}\); \(\left(\frac{x + y}{xy}\right)^{-1} = \frac{xy}{x + y}\).

Problem 2 Rules of exponents

Simplify.

a) \((3x^2y^{-3})^2\cdot(xy)^4\) b) \(\dfrac{12m^3n^{-2}}{4m^{-1}n^3}\) c) \(\dfrac{2^{n + 2} - 2^n}{3\cdot 2^{n - 1}}\) d) \(\dfrac{9^4\cdot 27^2}{3^{13}}\)

Show solution

a) \(9x^4y^{-6}\cdot x^4y^4 = \frac{9x^8}{y^2}\)
b) \(3m^4n^{-5} = \frac{3m^4}{n^5}\)
c) \(\frac{2^n(4 - 1)}{3\cdot 2^n/2} = 2\)
d) \(\frac{3^8\cdot 3^6}{3^{13}} = 3\)

Problem 3 Scientific notation

a) Write in scientific notation: 45 000 000; 0.000 072; 0.000 000 001 3.

b) Calculate \(\dfrac{360\,000\cdot 0.000\,05}{0.0012}\) using scientific notation.

Show solution

a) \(4.5\cdot 10^7\); \(7.2\cdot 10^{-5}\); \(1.3\cdot 10^{-9}\).

b) \(\frac{3.6\cdot 10^5\cdot 5\cdot 10^{-5}}{1.2\cdot 10^{-3}} = \frac{18}{1.2\cdot 10^{-3}} = 1.5\cdot 10^4\).

Problem 4 Exponential equations

Solve.

a) \(4^x = 32\) b) \(9^{x - 1} = 27\) c) \(\left(\frac15\right)^{x} = 125\) d) \((0.5)^{2x - 1} = 16\) e) \(5^x = 0.0016\)

Show solution

a) \(2^{2x} = 2^5\): \(x = \frac52\)
b) \(3^{2x - 2} = 3^3\): \(x = \frac52\)
c) \(5^{-x} = 5^3\): \(x = -3\)
d) \(2^{-(2x - 1)} = 2^4\): \(x = -\frac32\)
e) \(0.0016 = \frac{16}{10000} = \frac{1}{625} = 5^{-4}\): \(x = -4\)