Final answer:
To rationalize the denominator 1/(sqrt 5)(cuberoot 5), we multiply both numerator and denominator by a term that when combined with the denominator's terms, yields a whole number exponent for the base 5, thus eliminating the radicals and rationalizing the denominator.
Step-by-step explanation:
To rationalize the denominator 1/(sqrt 5)(cuberoot 5), we aim to eliminate the radicals from the denominator. We do this by multiplying both numerator and denominator by the appropriate terms that will result in a rational number in the denominator. Taking our cue from the power rules of exponents, we can say that sqrt 5 is equivalent to 5^(1/2), and cuberoot 5 is equivalent to 5^(1/3).
Knowing that when we multiply exponents with the same base we add the exponents, we will look for a common exponent that will give us a whole number when combined with both 1/2 and 1/3. This common exponent is the least common multiple of 2 and 3, which is 6. Hence, we will multiply both numerator and denominator by 5^(2/3) * 5^(1/2) to get a denominator of 5^(1), which is just 5. The process is analogous to using conversion factors where units in the numerator and denominator cancel out, paralleling how the radicals in our case will cancel out.
The result will be:
- Multiply the numerator by 5^(2/3) * 5^(1/2).
- Multiply the denominator by 5^(2/3) * 5^(1/2).
- Combine the exponents in the denominator to attain a rational number.
- Simplify by canceling equivalent terms in the numerator and denominator, if possible.