Sure, let's work through the expression step-by-step to find the value when [tex]\(a = 3\)[/tex] and [tex]\(b = -2\)[/tex].
The given expression is:
[tex]\[
\left(\frac{3 a^{-2} b^6}{2 a^{-1} b^5}\right)^2
\][/tex]
1. Substitute the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
- Let [tex]\(a = 3\)[/tex]
- Let [tex]\(b = -2\)[/tex]
2. Evaluate the numerator:
[tex]\[
3 \cdot (3^{-2}) \cdot ((-2)^6)
\][/tex]
- [tex]\(3^{-2}\)[/tex] is the same as [tex]\(\frac{1}{3^2} = \frac{1}{9}\)[/tex]
- [tex]\((-2)^6\)[/tex] is [tex]\(64\)[/tex]
Therefore, the numerator becomes:
[tex]\[
3 \cdot \frac{1}{9} \cdot 64 = \frac{3 \cdot 64}{9} = \frac{192}{9} = 21.333333333333332
\][/tex]
3. Evaluate the denominator:
[tex]\[
2 \cdot (3^{-1}) \cdot ((-2)^5)
\][/tex]
- [tex]\(3^{-1}\)[/tex] is the same as [tex]\(\frac{1}{3}\)[/tex]
- [tex]\((-2)^5\)[/tex] is [tex]\(-32\)[/tex]
Therefore, the denominator becomes:
[tex]\[
2 \cdot \frac{1}{3} \cdot (-32) = \frac{2 \cdot (-32)}{3} = \frac{-64}{3} = -21.333333333333332
\][/tex]
4. Divide the numerator by the denominator:
[tex]\[
\frac{21.333333333333332}{-21.333333333333332} = -1
\][/tex]
5. Square the result:
[tex]\[
(-1)^2 = 1
\][/tex]
So, the value of the expression [tex]\(\left(\frac{3 a^{-2} b^6}{2 a^{-1} b^5}\right)^2\)[/tex] when [tex]\(a = 3\)[/tex] and [tex]\(b = -2\)[/tex] is:
[tex]\[
1
\][/tex]
