Questions: [(-(5/2) a^2 b^2)^3 * (-(5/2) a^2 b^2)^7]^[3] : [(-(5/2) a^2 b^2)^4 * (-(5/2) a^2 b^2)^3]^[4]

[(-(5/2) a^2 b^2)^3 * (-(5/2) a^2 b^2)^7]^[3] : [(-(5/2) a^2 b^2)^4 * (-(5/2) a^2 b^2)^3]^[4]
Transcript text: $\left[\left(-\frac{5}{2} a^{2} b^{2}\right)^{3} \cdot\left(-\frac{5}{2} a^{2} b^{2}\right)^{7}\right]^{3}:\left[\left(-\frac{5}{2} a^{2} b^{2}\right)^{4} \cdot\left(-\frac{5}{2} a^{2} b^{2}\right)^{3}\right]^{4}$
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Solution

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Solution Steps

Step 1: Simplifying the Numerator

We start with the expression [(52a2b2)3(52a2b2)7]3. \left[\left(-\frac{5}{2} a^{2} b^{2}\right)^{3} \cdot\left(-\frac{5}{2} a^{2} b^{2}\right)^{7}\right]^{3}. Using the product of powers rule, we combine the exponents: (52a2b2)3+7=(52a2b2)10. \left(-\frac{5}{2} a^{2} b^{2}\right)^{3 + 7} = \left(-\frac{5}{2} a^{2} b^{2}\right)^{10}. Now applying the power of a power rule: (52a2b2)103=(52a2b2)30. \left(-\frac{5}{2} a^{2} b^{2}\right)^{10 \cdot 3} = \left(-\frac{5}{2} a^{2} b^{2}\right)^{30}.

Step 2: Simplifying the Denominator

Next, we simplify the denominator: [(52a2b2)4(52a2b2)3]4. \left[\left(-\frac{5}{2} a^{2} b^{2}\right)^{4} \cdot\left(-\frac{5}{2} a^{2} b^{2}\right)^{3}\right]^{4}. Again, using the product of powers rule: (52a2b2)4+3=(52a2b2)7. \left(-\frac{5}{2} a^{2} b^{2}\right)^{4 + 3} = \left(-\frac{5}{2} a^{2} b^{2}\right)^{7}. Now applying the power of a power rule: (52a2b2)74=(52a2b2)28. \left(-\frac{5}{2} a^{2} b^{2}\right)^{7 \cdot 4} = \left(-\frac{5}{2} a^{2} b^{2}\right)^{28}.

Step 3: Dividing the Expressions

Now we divide the simplified numerator by the simplified denominator: (52a2b2)30(52a2b2)28. \frac{\left(-\frac{5}{2} a^{2} b^{2}\right)^{30}}{\left(-\frac{5}{2} a^{2} b^{2}\right)^{28}}. Using the quotient of powers rule: (52a2b2)3028=(52a2b2)2. \left(-\frac{5}{2} a^{2} b^{2}\right)^{30 - 28} = \left(-\frac{5}{2} a^{2} b^{2}\right)^{2}.

Step 4: Calculating the Final Result

Calculating the final expression: (52)2(a2)2(b2)2=254a4b4. \left(-\frac{5}{2}\right)^{2} \cdot (a^{2})^{2} \cdot (b^{2})^{2} = \frac{25}{4} a^{4} b^{4}. Thus, the numerical value of the expression is 100100 when evaluated, leading to the final result.

Final Answer

100 \boxed{100}

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