Solve for x 9^(x^2)*3^(7x)=81

Problem

9^(x^2)⋅3^(7*x)=81

Solution

1. Rewrite all terms using a common base of 3

9=3^2

81=3^4

2. Substitute these values back into the original equation.

(3^2)^(x^2)⋅3^(7*x)=3^4

3. Apply the power of a power rule (a^m)^n=a^(m*n) to the first term.

3^(2*x^2)⋅3^(7*x)=3^4

4. Apply the product rule for exponents a^m⋅a^n=a^(m+n) to combine the terms on the left side.

3^(2*x^2+7*x)=3^4

5. Equate the exponents since the bases are equal.

2*x^2+7*x=4

6. Rearrange the equation into standard quadratic form a*x^2+b*x+c=0

2*x^2+7*x−4=0

7. Factor the quadratic expression.

(2*x−1)*(x+4)=0

8. Solve for x by setting each factor equal to zero.

2*x−1=0⇒x=1/2

x+4=0⇒x=−4

Final Answer

x=1/2,−4

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