Sure, let's work through this problem step by step.
We start with the expression:
[tex]\[ \sqrt[3]{\left(\frac{y}{x}\right)^{-5}} \][/tex]
Step 1: Simplify the exponent
The expression inside the cube root is:
[tex]\[ \left(\frac{y}{x}\right)^{-5} \][/tex]
When you have a negative exponent, you can rewrite it by taking the reciprocal of the base and changing the sign of the exponent:
[tex]\[ \left(\frac{y}{x}\right)^{-5} = \left(\frac{x}{y}\right)^{5} \][/tex]
Step 2: Take the cube root of the expression
Next, we need to take the cube root of [tex]\(\left(\frac{x}{y}\right)^{5}\)[/tex]:
[tex]\[ \sqrt[3]{\left(\frac{x}{y}\right)^{5}} \][/tex]
The cube root of an expression with an exponent [tex]\(a\)[/tex] is the same as raising the expression to the power of [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[ \left(\left(\frac{x}{y}\right)^{5}\right)^{\frac{1}{3}} \][/tex]
Step 3: Apply the power of a power rule
When raising a power to another power, you multiply the exponents:
[tex]\[ \left(\frac{x}{y}\right)^{5 \cdot \frac{1}{3}} \][/tex]
Simplifying the exponents:
[tex]\[ 5 \cdot \frac{1}{3} = \frac{5}{3} \][/tex]
So the expression becomes:
[tex]\[ \left(\frac{x}{y}\right)^{\frac{5}{3}} \][/tex]
Therefore, the simplified form of the original expression is:
[tex]\[ \left(\frac{x}{y}\right)^{\frac{5}{3}} \][/tex]
To approximate this exponent numerically, [tex]\(\frac{5}{3}\)[/tex] is approximately equal to 1.66666666666667, so the expression can also be written as:
[tex]\[ \left(\frac{x}{y}\right)^{1.66666666666667} \][/tex]
Thus, the result is:
[tex]\[ \boxed{\left(\frac{x}{y}\right)^{1.66666666666667}} \][/tex]
