Questions: Simplify the exponential expression. ((-28 a^5 b^8)/(7 a^8 b^-2))^3 ((-28 a^5 b^8)/(7 a^8 b^-2))^3 = (Simplify your answer. Use positive exponents only.)

Solution Steps

To simplify the given exponential expression, follow these steps:

1. Simplify the fraction inside the parentheses by dividing the coefficients and applying the laws of exponents to the variables.
2. Raise the simplified fraction to the power of 3 by applying the power of a quotient rule, which involves raising both the numerator and the denominator to the power of 3.
3. Ensure all exponents are positive by using the property \(a^{-n} = \frac{1}{a^n}\).

We start with the expression:

\[ \left(\frac{-28 a^{5} b^{8}}{7 a^{8} b^{-2}}\right)^{3} \]

First, we simplify the fraction inside the parentheses:

\[ \frac{-28}{7} = -4 \]

For the variables, we apply the laws of exponents:

\[ \frac{a^{5}}{a^{8}} = a^{5-8} = a^{-3} \] \[ \frac{b^{8}}{b^{-2}} = b^{8 - (-2)} = b^{8 + 2} = b^{10} \]

Thus, the simplified fraction becomes:

\[ \frac{-4 b^{10}}{a^{3}} \]

Next, we raise the simplified fraction to the power of 3:

\[ \left(-4 b^{10} a^{-3}\right)^{3} = (-4)^{3} (b^{10})^{3} (a^{-3})^{3} \]

Calculating each part:

\[ (-4)^{3} = -64 \] \[ (b^{10})^{3} = b^{30} \] \[ (a^{-3})^{3} = a^{-9} \]

Combining these results gives us:

\[ -64 b^{30} a^{-9} \]

To express the final result with positive exponents, we rewrite \(a^{-9}\) as \(\frac{1}{a^{9}}\):

\[ -64 \frac{b^{30}}{a^{9}} = \frac{-64 b^{30}}{a^{9}} \]

Final Answer