Question

x / (-3/2)^-3=(-8/27)^-2

Answer 1

Please find the attached image of the solution.

Answer 2

Hey there!

Here's the detailed process of my calculations:

Given equation:

$\bf{\mathsf{\frac{x}{(\frac{- 3}{2})^{- 3}} = (\frac{- 8}{27})^{- 2}}}$

By applying the fraction and exponential rules of equations in left hand side that is,

$\bf{\mathsf{\frac{- a}{b} = - \frac{a}{b} \: \: \: \: and \: \: \: (- a)^n = - a^n}}$

Now,

$\bf{\mathsf{- \frac{x}{(\frac{3}{2})^{- 3}} = (\frac{- 8}{27})^{- 2}}}$

Now, by applying the rules of exponents into this equation that is,

$\bf{\mathsf{(\frac{a}{b})^{- c} = ((\frac{a}{b})^{- 1})^c = (\frac{b}{a})^c}}$

And, by applying the exponential rule that is,

$\bf{\mathsf{(\frac{a}{b})^c = \frac{a^c}{b^c}}}$

Therefore,

$\bf{\mathsf{- \frac{x}{\frac{2^3}{3^3}} = (\frac{- 8}{27})^{- 2}}}$

Applying the rules of fraction into this that is,

$\bf{\mathsf{\frac{a}{\frac{b}{c}} = \frac{a \times c}{b}}}$

$\bf{\mathsf{- \frac{3^3 x}{2^3} = (\frac{- 8}{27})^{- 2}}}$

$\bf{\mathsf{- \frac{27x}{8} = (\frac{- 8}{27})^{- 2}}}$

Applying the fractional and exponential rules in Right hand side that is,

$\bf{\mathsf{1) \: \: \: \: \frac{- a}{b} = - \frac{a}{b}}}$

$\bf{\mathsf{2) \: \: \: \: a^{- b} = \frac{1}{a^b}}}$

$\bf{\mathsf{3) \: \: \: \: (- a)^n = a^n, \qquad If \: \: n \: \: is \: \: even.}}$

Hence,

$\bf{\mathsf{- \frac{27x}{8} = \frac{1}{(\frac{8}{27})^2}}}$

Applying the exponential rule that is,

$\bf{\mathsf{(\frac{a}{b})^c = \frac{a^c}{b^c}}}$

Apply the fraction rule that is,

$\bf{\mathsf{\frac{1}{\frac{b}{c}} = \frac{c}{b}}}$

Therefore,

$\bf{\mathsf{- \frac{27x}{8} = \frac{27^2}{8^2}}}$

$\bf{\mathsf{- \frac{27x}{8} = \frac{729}{64}}}$

$\bf{\mathsf{- 27x = \frac{729}{8}}}$

Divide both the sides by the value of "- 27" and simplify.

$\bf{\mathsf{\frac{- 27x}{- 27} = \frac{729}{\frac{8}{- 27}}}}$

$\bf{\mathsf{x = - \frac{729}{8 \times 27}}}$

$\bf{\mathsf{x = - \frac{729}{216}}}$

Cancel out the common term of "27" to obtain the final value.

$\bf{\mathsf{\boxed{\therefore \: \: x = - \frac{27}{8}}}}$

Which is the required solution for this type of query.

Hope this extremely detailed answer helps you and clears the rules to be used simultaneously!