Problem 100 - College Algebra

Question

Simplify using properties of exponents. $$\frac{\left(2 y^{\frac{1}{5}}\right)^{4}}{y^{\frac{3}{10}}}$$

Step-by-Step Solution

Answer

The simplified version of the given expression is \(16y^{\frac{1}{2}}\).

Step 1: Simplify Inner Exponent

Start by simplifying the inner exponent where \(2y^{\frac{1}{5}}\) is raised to power of 4. The expression turns into \(2^4 * (y^{\frac{1}{5}})^4\) which simplifies to \(16 * y^{\frac{4}{5}}\) as when raising an exponent to another exponent, the exponents are multiplied.

Step 2: Divide the Exponents

To divide exponents with the same base, the rule is to subtract the exponent in the denominator from the exponent in the numerator. So the expression \( \frac{16y^{\frac{4}{5}}}{y^{\frac{3}{10}}}\) turns into \(16 * y^{(\frac{4}{5})-(\frac{3}{10})}\) .

Step 3: Subtract the Exponents

Subtracting the exponents, maintaining a common denominator to make calculations easy, yields \(16y^{(\frac{8}{10} - \frac{3}{10})}\). This simplifies to \(16y^{\frac{1}{2}}\).