The expression start root, start index, 4, end index, 6, to the power 5 , end root, dot, square root of, 6, cubed, end square root

Question

4

6
5

​
⋅
6
3

​
is equivalent to
Answer
Multiple Choice Answers
6, start superscript, start fraction, 15, divided by, 8, end fraction, end superscript6
8
15
​


6, start superscript, start fraction, 8, divided by, 15, end fraction, end superscript6
15
8
​


6, start superscript, start fraction, 4, divided by, 11, end fraction, end superscript6
11
4
​


6, start superscript, start fraction, 11, divided by, 4, end fraction, end superscript6
4
11
​

Answer 1

To simplify the expression \( \sqrt[4]{6^5} \cdot \sqrt{6^3} \), we can break it down into steps.

1. Express the radicals in exponent form:

   • The fourth root of \(6^5\) can be rewritten as: \[ \sqrt[4]{6^5} = 6^{5/4} \]

   • The square root of \(6^3\) can be rewritten as: \[ \sqrt{6^3} = 6^{3/2} \]

2. Combine the exponents: Now we can combine the two exponents: \[ 6^{5/4} \cdot 6^{3/2} \] To combine, we add the exponents: \[ 5/4 + 3/2 \]

   To add these, we need a common denominator, which in this case is 4: \[ 3/2 = 6/4 \quad \text{(since \(3 \times 2 = 6\) and \(2 \times 2 = 4\))} \] Now we can add: \[ 5/4 + 6/4 = (5 + 6)/4 = 11/4 \]

3. Final expression: Hence, we can express the result as: \[ 6^{11/4} \]

Now we check the equivalent options provided:

• \( 6^{\frac{15}{8}} \)
• \( 6^{\frac{8}{15}} \)
• \( 6^{\frac{4}{11}} \)
• \( 6^{\frac{11}{4}} \)

The correct equivalent expression is:

\[ \boxed{6^{\frac{11}{4}}} \]

This matches the last option, which is \( 6^{\frac{11}{4}} \).