Problem 99 - College Algebra

Question

Simplify using properties of exponents. $$\frac{\left(3 y^{\frac{1}{4}}\right)^{3}}{y^{\frac{1}{12}}}$$

Step-by-Step Solution

Answer

The simplified form of \(\frac{(3 y^{1/4})^{3}}{y^{1/12}}\) is \(27 y^{2/3}\).

Step 1: Rewrite the power of a power expression

To simplify, the first step is to rewrite the power of a power expression on the numerator by multiplying the exponents. Given \((3 y^{1/4})^3\), apply the rule \(a^{m \cdot n} = (a^m)^n\) to get, \(3^3 y^{(1/4) \cdot 3} = 27 y^{3/4}\).

Step 2: Subtract the exponents

The second step is to subtract the exponent of the denominator from the exponent of the numerator. Given \(27 y^{3/4} / y^{1/12}\), apply the rule \(a^{m}/a^{n} = a^{m-n}\) to get, \(27 y^{(3/4 - 1/12)}\).

Step 3: Simplify the exponent

Convert both fractions to a common denominator and calculate the difference. From step 2, simplify the exponent of \(y\) in \(27 y^{(3/4 - 1/12)}\) to \(27 y^{(9/12 - 1/12) = 27 y^{8/12}}.\); this can be further simplified to \(27 y^{2/3}.\)