Evaluate (27^(-2/3))/(27^(-1/3))

Problem

(27(−2/3))/(27(−1/3))

Solution

1. Apply the quotient rule for exponents, which states that (am)/(an)=a(m−n)

2. Subtract the exponents by taking the exponent of the numerator and subtracting the exponent of the denominator.

(27(−2/3))/(27(−1/3))=27(−2/3−(−1/3))

3. Simplify the exponent by performing the addition of the fractions.

−2/3+1/3=−1/3

4. Rewrite the expression using the simplified exponent.

27(−1/3)

5. Apply the negative exponent rule a(−n)=1/(an) to move the base to the denominator.

1/(27(1/3))

6. Evaluate the cube root since a(1/3)=√(3,a)

√(3,27)=3

7. Simplify the final fraction using the result of the cube root.

1/3

Final Answer

(27(−2/3))/(27(−1/3))=1/3

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