Verified Answer :
Answer:
\(x=\frac{-3}{2}\)
Explanation:
We have been given with the expression \((1/3)^x=(27)^x+2\)
Now, to solve the equation firstly we have to make the base same on both sides
\((1/3)^x=3^{-x}\)
27 can be written as \(3^3\)
\(27^x=(3^3)^x=3^{3x}\)
Hence, given expression can be rewritten as
\(3^{-x}=3^{3(x+2)}\)
Now since, base is same we can equate the powers on both sides
\(-x=3(x+2)\\ \Rightarrow-x= 3x+6\\ \Rightarrow -x-3x=6\\-4x=6\)\(\Rightarrow x=\frac{-3}{2}\)
Therefore given expression \((1/3)^x=(27)^x+2\) is equivalent to \(x=\frac{-3}{2}\)
Equivalent means the simplified form of any given expression
To find the equivalent expression, we need to expand the given expression by using law of indices.
The equation \(3^{-x} = 3^{3x} + 2\) is equivalent to \(\left (\dfrac {1}{3}\right )^x = 27 ^x + 2\).
Given:
The given equation is \(\left (\dfrac {1}{3}\right )^x = 27 ^x + 2\).
Solving left hand side of the equation.
\(\left (\dfrac {1}{3}\right )^x\)
Apply the law of indices.
\(\left (\dfrac {1}{3}\right )^x=(-3)^x\)
Solving right hand side of the equation.
\(27 ^x + 2=(3^{3x}+2)\)
Thus, the equation \(3^{-x} = 3^{3x} + 2\) is equivalent to \(\left (\dfrac {1}{3}\right )^x = 27 ^x + 2\).
Learn more about law of indices here:
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