How do I solve for x in the equation, 9^x-2×81^3x/2=3^2(10x)?

To solve for x in the equation 9^(x – 2) * 81^(3x/2) = 3^(2(10x)), we need to simplify and manipulate the equation step by step.

Let’s break it down:

Step 1: Rewrite 81 as 3^4 since 81 is equal to 3^4. So, the equation becomes: 9^(x – 2) * (3^4)^(3x/2) = 3^(2(10x))

Step 2: Use the properties of exponents to simplify further. When you raise a power to another power, you multiply the exponents. 9^(x – 2) * 3^(4(3x/2)) = 3^(2(10x))

Step 3: Now, apply the distributive property of multiplication with exponents on the left side: 9^(x – 2) * 3^(6x) = 3^(2(10x))

Step 4: Notice that both sides have a base of 3. To equate the exponents, we can set the exponents equal to each other: x – 2 + 6x = 2(10x)

Step 5: Simplify the equation by combining like terms: 7x – 2 = 20x

Step 6: Move the 7x term to the other side of the equation by subtracting 7x from both sides: -2 = 13x

Step 7: Finally, solve for x by dividing both sides by 13: x = -2/13

So, the solution to the equation 9^(x – 2) * 81^(3x/2) = 3^(2(10x)) is x = -2/13.

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