
Answers to Questions on Power Functions
 Question:
Evaluate 8^{1/3} and 8^{1/3}
Answer: 2, 1/2
 Question: Evaluate 8^{2/3} and
8^{2/3}
Answer: 4, 1/4
 Question: Evaluate 4^{5/2},
27^{4/3} and 16^{3/2}
Answer: 32, 81, 1/64
Return to Exercises
 Question:
In the table below you can see the values of three different functions.
Two are power functions: one has the form f(t) = at^{2}, while
the other has the form f(t) = bt^{3}. The third is an
exponential
function of the form f(t) = kb^{t}. Which is which and how can
you tell?
t  F_{1}(t)  t  F_{2}(t)  t  F_{3}(t) 
1.0  2.5  0  2.2  2.0  4.8 
1.2  4.32  1.0  2.64  2.2  5.81 
1.4  6.86  2.0  3.17  2.4  6.91 
1.6  10.24  3.0  3.80  2.6  8.11 
1.8  14.58  4.0  4.56  2.8  9.41 
2.0  20.0  5.0  5.47  3.0  10.8 
Answer: F_{1}(t) is the
cubic function;
F_{2}(t) is the exponential function;
F_{3}(t) is the quadratic
function.
Here are some observations which lead to these conclusions.
 All power functions kt^{p} with p 0 are zero for t = 0.
Thus F_{2} cannot be a power function.
 Assuming that F_{1}(t) = kt^{n} with n = 2 or n = 3, we can evaluate F_{1}(1) to obtain k. In this case, k = 2.5.
 Since F_{1}(t) = (2.5)t^{n} 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.
Return to Exercises
 Question: Solve 3 x^{2.2} = 6.
Answer: x = 2^{1/(2.2)}. Since 2.2 = 22/10 = 11/5, this is the same as x = 2^{5/11}.
 Question: 3 x^{2.2} = 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 welldefined 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!
 Question: x^{ 4} = 1.7
Answer: x = (1.7)^{ 1/4}.
 Question: x^{ 4} = 1.7
Answer: x^{ 4} = 1/x^{4} which is never negative for real numbers x, so this equation has no solution in any interpretation.
