Solve for x ( square root of 3)^(x+1)=9^x

Problem

(√(,3))(x+1)=9

Solution

1. Rewrite the square root as a fractional exponent using the property √(,a)=a(1/2)

(√(,3))(x+1)=(3(1/2))(x+1)

2. Express the base 9 as a power of 3 to ensure both sides of the equation have the same base.

9=(3)x

3. Apply the power of a power rule (am)n=a(m*n) to simplify the exponents on both sides.

3(1/2*(x+1))=3(2*x)

4. Equate the exponents since the bases are now identical.

1/2*(x+1)=2*x

5. Eliminate the fraction by multiplying both sides of the equation by 2.

x+1=4*x

6. Isolate the variable by subtracting x from both sides.

1=3*x

7. Solve for x by dividing both sides by 3.

x=1/3

Final Answer

(√(,3))(x+1)=9⇒x=1/3

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