Given [tex]$x\ \textgreater \ 0$[/tex], the expression [tex]$\sqrt[6]{x^{65}}$[/tex] is equivalent to:
A. [tex]$x^{10} \sqrt[6]{x^6}$[/tex]
B. [tex]$x^9 \sqrt[6]{x^6}$[/tex]
C. [tex]$x^9 \sqrt[6]{x^5}$[/tex]
D. [tex]$x^{10} \sqrt[6]{x^5}$[/tex]
Answer (2)
Rewrite the expression using fractional exponents: 6 x 65 = x 6 65 .
Divide the exponent: 6 65 = 10 + 6 5 .
Substitute back into the expression: x 6 65 = x 10 + 6 5 = x 10 ⋅ x 6 5 .
Rewrite as a radical: x 10 ⋅ x 6 5 = x 10 6 x 5 .
The equivalent expression is x 10 6 x 5 .
Explanation
Understanding the Problem We are given the expression 6 x 65 where 0"> x > 0 , and we want to find an equivalent expression from the given options.
Rewriting with Fractional Exponents First, let's rewrite the expression using fractional exponents: 6 x 65 = x 6 65 .
Dividing the Exponent Now, we want to divide 65 by 6. We can express 65 as 65 = 6 ⋅ 10 + 5 . Therefore, 6 65 = 10 + 6 5 .
Substituting Back Substitute this back into the expression: x 6 65 = x 10 + 6 5 = x 10 ⋅ x 6 5 .
Rewriting as a Radical Finally, rewrite the fractional exponent as a radical: x 10 ⋅ x 6 5 = x 10 ⋅ 6 x 5 .
Final Answer Therefore, the expression 6 x 65 is equivalent to x 10 6 x 5 .
Examples
Understanding how to simplify radical expressions with fractional exponents is useful in various fields, such as physics and engineering, where you might encounter complex equations involving roots and powers. For example, when dealing with wave functions in quantum mechanics or analyzing stress distributions in mechanical structures, simplifying expressions with radicals and exponents can make calculations more manageable and provide clearer insights into the underlying phenomena. This skill helps in converting complex mathematical models into simpler, solvable forms, facilitating analysis and problem-solving.
The expression 6 x 65 can be simplified to x 10 6 x 5 by using fractional exponents. Therefore, the correct choice among the given options is x 10 6 x 5 .
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