Simplify square root of a^-2 cube root of a^5b^-3

Problem

√(,a(−2)√(3,a5*b(−3)))

Solution

1. Rewrite the radical expressions as fractional exponents to make manipulation easier.

√(,a(−2)*(a5*b(−3))(1/3))

2. Apply the power of a product rule (x*y)n=xn*yn to the inner term.

√(,a(−2)*a(5/3)*b(−3/3))

3. Simplify the exponents inside the square root, noting that −3/3=−1

√(,a(−2)*a(5/3)*b(−1))

4. Combine the terms with the base a by adding their exponents: −2+5/3=−6/3+5/3=−1/3

√(,a(−1/3)*b(−1))

5. Convert the outer square root into a fractional exponent of 1/2

(a(−1/3)*b(−1))(1/2)

6. Distribute the outer exponent to each term inside the parentheses by multiplying the exponents.

a(−1/6)*b(−1/2)

7. Express the result using positive exponents by moving the terms to the denominator.

1/(a(1/6)*b(1/2))

Final Answer

√(,a(−2)√(3,a5*b(−3)))=1/(a(1/6)*b(1/2))

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