QuestionSolve $\frac{36^{p-2}}{216^{q+2 p}}=1$ for $q$ in terms of $p$ .

Studdy Solution

STEP 1

Assumptions1. The given equation is $\frac{36^{p-}}{216^{q+ p}}=1$ . We need to express $q$ in terms of $p$

STEP 2

First, we need to simplify the equation. We can do this by expressing36 and216 as powers of6. $36 =6^2$ 216 =6^So, the equation becomes\frac{(6^2)^{p-2}}{(6^)^{q+2 p}}=1

STEP 3

Next, we simplify the equation further by applying the power of a power rule, which states that $(a^{m})^{n} = a^{mn}$ .
$\frac{6^{2(p-2)}}{6^{3*(q+2 p)}}=1$

STEP 4

implify the equation by multiplying the powers.
$\frac{6^{2p-4}}{6^{3q+6p}}=1$

STEP 5

Since the bases are equal, we can equate the exponents.
$2p-4 =3q+p$

STEP 6

Rearrange the equation to express $q$ in terms of $p$ .
$3q =2p -6p -4$

STEP 7

implify the equation to get the final expression for $q$ .
$q = \frac{2p -6p -4}{3} = \frac{-4p -4}{3}$ So, $q$ in terms of $p$ is $q = \frac{-4p -4}{3}$ .

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