Simplify ((16x^-10)/(x^6))^(-1/4)

Problem

((16*x^(−10))/(x^6))^(−1/4)

Solution

1. Simplify the expression inside the parentheses by using the quotient rule for exponents, a^m/a^n=a^(m−n)

(16*x^(−10))/(x^6)=16*x^(−10−6)

16*x^(−10−6)=16*x^(−16)

2. Apply the power of a product rule (a*b)^n=a^n*b^n to the simplified expression.

(16*x^(−16))^(−1/4)=16^(−1/4)⋅(x^(−16))^(−1/4)

3. Evaluate the numerical part by rewriting 16 as a power of 2 and applying the power of a power rule (a^m)^n=a^(m*n)

16^(−1/4)=(2^4)^(−1/4)

(2^4)^(−1/4)=2^(−1)

2^(−1)=1/2

4. Evaluate the variable part by multiplying the exponents.

(x^(−16))^(−1/4)=x^(−16⋅−1/4)

x^(−16⋅−1/4)=x^4

5. Combine the results to find the final simplified form.

1/2⋅x^4=(x^4)/2

Final Answer

((16*x^(−10))/(x^6))^(−1/4)=(x^4)/2

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