# Answers to Questions on Power Functions

1. Question: Evaluate 81/3 and 8-1/3

2. Question: Evaluate 82/3 and 8-2/3

3. Question: Evaluate 45/2, 274/3 and 16-3/2

Answer: 32, 81, 1/64

1. Question: In the table below you can see the values of three different functions. Two are power functions: one has the form f(t) = at2, while the other has the form f(t) = bt3. The third is an exponential function of the form f(t) = kbt. Which is which and how can you tell?

tF1(t)tF2(t)tF3(t)
1.02.502.22.04.8
1.24.321.02.642.25.81
1.46.862.03.172.46.91
1.610.243.03.802.68.11
1.814.584.04.562.89.41
2.020.05.05.473.010.8

F1(t) is the cubic function;

F2(t) is the exponential function;

F3(t) is the quadratic function.

Here are some observations which lead to these conclusions.

• All power functions ktp with p 0 are zero for t = 0. Thus F2 cannot be a power function.

• Assuming that F1(t) = ktn with n = 2 or n = 3, we can evaluate F1(1) to obtain k. In this case, k = 2.5.

• Since F1(t) = (2.5)tn with n = 2 or n = 3, we can figure out n by evaluating at t = 1.2.

(2.5)(1.2)2 = 3.6 and (2.5)(1.2)3 = 4.32,

so n = 3.

1. Question: Solve 3 x2.2 = 6.

Answer: x = 21/(2.2). Since 2.2 = 22/10 = 11/5, this is the same as x = 25/11.

2. Question: 3 x2.2 = -6.

Answer: x = (-2)1/(2.2). Since 2.2 = 22/10 = 11/5, this is the same as x = (-2)5/11 which is a well-defined real number. On the other hand, your calculator or computer algebra package will probably treat the exponent 1/(2.2) as a REAL number and, not caring that this is a rational number with an odd denominator, will feel that (-2)1/(2.2) is a complex number which is NOT real. You be the judge!

3. Question: x -4 = 1.7

Answer: x = (1.7) -1/4.

4. Question: x -4 = -1.7

Answer: x -4 = 1/x4 which is never negative for real numbers x, so this equation has no solution in any interpretation.

 Power Functions Field Guide HUB CQ Directory CQ Resources