Question

If 3 to the power of -3x + 6 equals 27 to the power of x + 4, then x equals __________.

Answer 1

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If 3 to the power of -3x + 6 equals 27 to the power of x + 4, then x equals _-1_ .

Explanation:

Let's solve the equation:

`\[3^(-3x + 6) = 27^(x + 4)\]`

First, express 27 as a power of 3, since both sides of the equation should have the same base:

`\[27 = 3^3\]`

Now substitute this into the equation:

`\[3^(-3x + 6) = (3^3)^(x + 4)\]`

Use the property
`\(a^(mn) = (a^m)^n\):`

`\[3^(-3x + 6) = 3^(3(x + 4))\]`

Now, set the exponents equal to each other:

`\[-3x + 6 = 3(x + 4)\]`

Expand and solve for x:

`\[-3x + 6 = 3x + 12\]`

Combine like terms:

`\[6 = 6x + 12\]`

Subtract 12 from both sides:

`\[-6 = 6x\]`

Divide by 6:

`\[x = -1\]`

Therefore,
`\(x\) equals \(-1\).`