$243^{-y}=\left(\frac{1}{243}\right)^{3 y} \cdot 9^{-2 y}$

Solve:
y=-1
y=0
y=1
no solution

Express all terms as powers of 3: 243 = 3 5 , 243 1 ​ = 3 − 5 , 9 = 3 2 .
Substitute into the original equation and simplify exponents: 3 − 5 y = 3 − 15 y ⋅ 3 − 4 y .
Combine terms: 3 − 5 y = 3 − 19 y .
Equate exponents and solve for y : − 5 y = − 19 y ⟹ y = 0 .

Explanation

Problem Analysis We are given the equation 24 3 − y = ( 243 1 ​ ) 3 y ⋅ 9 − 2 y and asked to solve for y . We will rewrite the equation using properties of exponents to find the solution.

Expressing as Powers of 3 First, express all terms as powers of 3. We know that 243 = 3 5 , 243 1 ​ = 3 − 5 , and 9 = 3 2 . Substituting these into the original equation, we get: ( 3 5 ) − y = ( 3 − 5 ) 3 y ⋅ ( 3 2 ) − 2 y

Simplifying Exponents Next, simplify the exponents using the property ( a b ) c = a b c : 3 − 5 y = 3 − 15 y ⋅ 3 − 4 y

Combining Terms Now, combine the terms on the right side using the property a b ⋅ a c = a b + c : 3 − 5 y = 3 − 15 y − 4 y
3 − 5 y = 3 − 19 y

Equating Exponents Since the bases are equal, we can equate the exponents: − 5 y = − 19 y

Solving for y Solve for y : − 5 y + 19 y = 0
14 y = 0
y = 0

Final Answer Therefore, the solution to the equation is y = 0 .


Examples
Imagine you're adjusting the settings on a sound equalizer. The equation we solved is similar to how audio engineers manipulate sound frequencies to achieve a balanced output. By understanding exponential relationships, they can fine-tune the amplitudes of different frequencies, ensuring the final sound is clear and harmonious. This involves adjusting parameters (like 'y' in our equation) to balance the overall sound profile, much like balancing exponents to solve for a variable.

Answered by GinnyAnswer | 2025-07-07