A rational exponent is written as xba, with base x and exponent ba. This expression is equivalent to the bth-root of x raised to the ath-power, or bxa. For example, 832could be rewritten as 382.
Note that the exponent is applied first, before the radical, and also that if the base is negative, taking roots is no longer simple, and requires Complex Number Exponentiation.
When solving questions like this, it is often easier to try to first represent the number inside the radical as an exponent. For example, 8 can be rewritten as 23. So,
What is a in the following equation: 234=a16?
Since 234=324=316, equating this with a16 gives a=3.□
We have 12532=212532=12521×32=(53)21×32=(53)31=53×31=5.□
What is x in the following equation: 532=(35)x?
Since 532=(531)2=(351)2=(35)2, equating this with (35)x gives x=2.□
We have 27−32=(33)−32=33×(−32)=3−2=(32)−1=321=91.□
What is a in the following equation: 3852=(a3)151?
Since 3852=831×52=8152=(82)151=(64)151=(43)151. Equating this with (a3)151 gives a=4.□
Which of the following is NOT equal to 8:(a)423(b)(2)6(c)43(d)6431?
Since xny=nxy, thus 423=243=43=64=8. This implies that neither of (a) and (c) is the answer. Now, since xny=(xn1)y=(nx)y=nxy, thus (2)6=26=64=8, implying that (b) is not the answer, either. Finally, since 6431=(26)31=26×31=22=4=8, thus the answer is (d).□
We might be tempted to simplify the exponents using the power rule, so that we would say ((−8)2)23=(−8)2×23=(−8)3=−512. However, this is incorrect!
We have to remember that order of operations requires us to perform operations inside parentheses before all other operations. Thus, the correct solution to this problem is as follows:
Note on Calculators
When entering an expression like the following into a calculator,
the results may or may not be correct, depending on how your calculator functions. The correct answer is 9, but some calculators may return an error or complex number as a result of shortcuts in the calculator's approach to rational exponents. Be careful when using your calculator on these sorts of problems.