QuestionSolve: $243^{-y}=\left(\frac{1}{243}\right)^{3 y} \cdot 9^{-2 y}$ $y=-1$ $y=0$ $y=1$ no solution

Studdy Solution

STEP 1

What is this asking? We need to find the value of $y$ that makes this equation with exponents true! Watch out! Exponent rules can be tricky, so let's be super careful with how we apply them.
Don't forget about negative exponents!

STEP 2

1. Rewrite the Equation
2. Simplify Using Exponent Rules
3. Solve for $y$

STEP 3

Alright, let's rewrite everything in terms of base 3!
We know that $243$ is $3^5$ and $9$ is $3^2$ , so let's substitute those in: $(3^5)^{-y} = \left(\frac{1}{3^5}\right)^{3y} \cdot (3^2)^{-2y}$

STEP 4

Remember, a fraction is just a negative exponent!
So $\frac{1}{3^5}$ is the same as $3^{-5}$ .
Let's replace that: $(3^5)^{-y} = (3^{-5})^{3y} \cdot (3^2)^{-2y}$

STEP 5

Now, let's use the power rule!
When you raise a power to a power, you multiply the exponents.
So, let's multiply those exponents: $3^{5 \cdot (-y)} = 3^{-5 \cdot 3y} \cdot 3^{2 \cdot (-2y)}$ $3^{-5y} = 3^{-15y} \cdot 3^{-4y}$

STEP 6

When we multiply terms with the same base, we add the exponents.
Let's do that on the right side: $3^{-5y} = 3^{-15y + (-4y)}$ $3^{-5y} = 3^{-19y}$

STEP 7

Since the bases are the same, the exponents must be equal!
So, we can set the exponents equal to each other: $-5y = -19y$

STEP 8

Let's add $19y$ to both sides to get all the $y$ terms on one side: $-5y + 19y = -19y + 19y$ $14y = 0$

STEP 9

Finally, let's divide both sides by $14$ to solve for $y$ : $\frac{14y}{14} = \frac{0}{14}$ $y = 0$

STEP 10

So, the value of $y$ that makes the equation true is $y = 0$ !

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