Math

QuestionSolve for xx in the equation (3x?)2=9x6(3 x^{?})^{2}=\frac{9}{x^{6}}.

Studdy Solution

STEP 1

Assumptions1. The expression is (3x?)=9x6\left(3 x^{?}\right)^{}=\frac{9}{x^{6}} . We are solving for the unknown exponent in the base x of the left-hand side of the equation.
3. The base x is the same in both sides of the equation.

STEP 2

First, we need to simplify the left-hand side of the equation. We can do this by applying the power of a power rule, which states that (am)n=amn(a^m)^n = a^{mn}.
(x?)2=2(x?)2\left( x^{?}\right)^{2} =^{2} \cdot \left(x^{?}\right)^{2}

STEP 3

Calculate the square of3.
32=93^{2} =9So, the left-hand side of the equation becomes9(x?)29 \cdot \left(x^{?}\right)^{2}

STEP 4

Now, we can rewrite the right-hand side of the equation to have the same base as the left-hand side. We can do this by rewriting 9x6\frac{9}{x^{6}} as 9x69 \cdot x^{-6}.
9x6=9x6\frac{9}{x^{6}} =9 \cdot x^{-6}

STEP 5

Now we equate the simplified left-hand side and right-hand side of the equation.
9(x?)2=9x9 \cdot \left(x^{?}\right)^{2} =9 \cdot x^{-}

STEP 6

We can now remove the common factor9 from both sides of the equation.
(x?)2=x6\left(x^{?}\right)^{2} = x^{-6}

STEP 7

Applying the power of a power rule again, we getx2?=x6x^{2?} = x^{-6}

STEP 8

Since the bases are the same, we can equate the exponents.
2?=62? = -6

STEP 9

Finally, we solve for the unknown ?.
?=6/2? = -6 /2

STEP 10

Calculate the value of ?.
?=6/2=3? = -6 /2 = -3The unknown exponent is -3.

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