Solve for x in the equation 3^(x^2)/9^x = 81

Published on October 31, 2024

Algebra

The equation 3^(x^2)/9^x = 81 is solved by rewriting the equation with the same base, simplifying the denominator, applying the quotient rule for exponents, equating the exponents, rearranging into a quadratic equation, and finally using the quadratic formula to find the two solutions, x = 1 + √5 and x = 1 - √5.

Question

\( \begin{array}{c}\frac{3^{x^{2}}}{9^{x}}=81 \\ x=?\end{array} \)

Answer

Here’s how to solve the equation:

1. Rewrite the equation using the same base:

The equation is

. Notice that 9 and 81 can be expressed as powers of 3:

and

. Substituting these into the equation gives us:

2. Simplify the denominator:

Using the power of a power rule,

, we simplify the denominator:

3. Use the quotient rule for exponents:

The quotient rule states that

. Applying this rule to our equation gives:

4. Equate the exponents:

Since the bases are the same, the exponents must be equal:

5. Solve the quadratic equation:

Rearrange the equation into standard quadratic form:

This quadratic equation doesn’t factor easily, so we’ll use the quadratic formula:

, where

, and

Therefore, the solutions are

and

Final Answer: The final answer is

🚀 Try Free Math Help Now!🏠 Learn More Powerful Features

Related math solutions

Simplifying Radical Expressions and Evaluating Algebraic Expressions

Step-by-step solutions for simplifying complex radical expressions and evaluating algebraic polynomials with square roots based on math textbook problems.

Solve an Absolute Value Equation with Fractions

Solve |(5x−1)/2| = |(3x+1)/4| by multiplying both sides by 4, then considering equal and opposite expressions inside the absolute values. The solutions are x = 3/7 and x = 1/13.

Identifying Systems of Inequalities from Graphs: Parabola and Absolute Value

This problem involves determining the correct system of inequalities based on a graph showing a shaded region bounded by a solid parabola and a dashed absolute value function. The solution requires analyzing the line types (solid vs. dashed) to determine strict or non-strict inequalities and identifying the equations of the boundary curves.

Comprehensive Mathematics Exam Solutions for Trigonometry, Algebra, and Calculus

Step-by-step solutions to a variety of mathematical problems including trigonometric equations, joint variation, quadratic equations, sequence formulas, and proofs.

Finding Reflections Across the Axis of Symmetry

Learn how to find the reflections of given coordinate points across the vertical axis of symmetry x = -3 using horizontal distance formulas.

🔎 Explore All Shared Math Solutions