Write the expression with rational​ exponents, and then apply the properties of exponents. Assume that all radicands represent positive real numbers. Give your answer in exponential form. 3√5√y

To rewrite the expression \(3 \sqrt{5 \sqrt{y}}\) using rational exponents and simplify, we can proceed as follows:

1. Rewrite the square root as a rational exponent:  
   \(\sqrt{y} = y^{\frac{1}{2}}\), so the expression inside becomes \(5 y^{\frac{1}{2}}\).

2. Now we have \(3 \sqrt{5 y^{\frac{1}{2}}}\). We can rewrite this outer square root as well:
   \[
   3 \sqrt{5 y^{\frac{1}{2}}} = 3 \cdot (5 y^{\frac{1}{2}})^{\frac{1}{2}}
   \]

3. Apply the exponent to both terms inside the parentheses:
   \[
   = 3 \cdot 5^{\frac{1}{2}} \cdot \left(y^{\frac{1}{2}}\right)^{\frac{1}{2}}
   \]

4. Simplify the exponent on \(y\) by multiplying the exponents:
   \[
   = 3 \cdot 5^{\frac{1}{2}} \cdot y^{\frac{1}{4}}
   \]

Thus, the simplified expression in exponential form is:
\[
3 \cdot 5^{\frac{1}{2}} \cdot y^{\frac{1}{4}}
\]

Would you like further clarification or details on any part of this solution?

Here are 5 related questions to deepen your understanding:

1. How do you rewrite cube roots using rational exponents?
2. What are the properties of exponents when multiplying terms with the same base?
3. How do you simplify expressions with fractional exponents?
4. How do you handle expressions with nested radicals?
5. How do you convert between radical form and exponential form?

**Tip:** Remember, \(x^{\frac{m}{n}}\) represents the \(n\)-th root of \(x\) raised to the \(m\)-th power: \(x^{\frac{m}{n}} = \sqrt[n]{x^m}\).

Write the expression with rational exponents, and then apply the properties of exponents. Assume that all radicands represent positive real numbers. Give your answer in exponential form. 3√5√y

Learn how to rewrite and simplify expressions with rational exponents through this example of 3√5√y. This detailed solution covers the application of exponent properties and converting nested radicals into exponential form, suitable for students in Grades 10-12.

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