To solve the problem, let's denote:
[tex]\[ a = \left(\frac{-3}{2}\right)^{-3} \][/tex]
[tex]\[ b = \left(\frac{9}{8}\right)^{-2} \][/tex]
We are asked to find the number [tex]\( x \)[/tex] such that:
[tex]\[ a \times x = b \][/tex]
Firstly, let's calculate the value of [tex]\( a \)[/tex]:
[tex]\[ a = \left(\frac{-3}{2}\right)^{-3} \][/tex]
The negative exponent rule states that:
[tex]\[ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n} \][/tex]
Thus,
[tex]\[ a = \left(\frac{-3}{2}\right)^{-3} = \left(\frac{2}{-3}\right)^{3} = \left(\frac{2}{-3}\right)^{3} \][/tex]
[tex]\[ = \frac{2^3}{(-3)^3} = \frac{8}{-27} = -\frac{8}{27} \][/tex]
To convert this to a decimal:
[tex]\[ a \approx -0.2962962962962963 \][/tex]
Next, let's calculate the value of [tex]\( b \)[/tex]:
[tex]\[ b = \left(\frac{9}{8}\right)^{-2} \][/tex]
By the same rule,
[tex]\[ b = \left(\frac{8}{9}\right)^{2} \][/tex]
[tex]\[ = \frac{8^2}{9^2} = \frac{64}{81} \][/tex]
To convert this to a decimal:
[tex]\[ b \approx 0.7901234567901234 \][/tex]
Now we need to solve for [tex]\( x \)[/tex] in the equation [tex]\( a \times x = b \)[/tex]:
[tex]\[ x = \frac{b}{a} \][/tex]
Substituting in the values we calculated for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = \frac{0.7901234567901234}{-0.2962962962962963} \][/tex]
Performing the division:
[tex]\[ x \approx -2.6666666666666665 \][/tex]
Hence, the number by which [tex]\(\left(\frac{-3}{2}\right)^{-3}\)[/tex] should be multiplied in order to get [tex]\(\left(\frac{9}{8}\right)^{-2}\)[/tex] is approximately [tex]\(-2.67\)[/tex].
