Solving the Equation: 27^x = 9/3^x: A Step-by-Step Guide! How To Solve An Exponential Equation With Unknowns On Both Sides! A Nice Exponential Algebra Problem

Algebraic equations with exponents can be tricky, especially when they involve different bases like 27^x = 9/3^x. Whether you're preparing for a math test, brushing up on your skills, or just love solving equations, understanding how to tackle these kinds of exponential equations is essential. In this blog post, we’ll break down the solution to this equation into easy-to-follow steps. Get ready to unravel the mystery behind 27^x =9/3^x and boost your math confidence!

 

Step 1: Rewrite the Equation Using the Same Base

The key to solving exponential equations is expressing the numbers with the same base whenever possible. Let's start by rewriting 27 and 9 as powers of 3.

27 = 3^3,   9 = 3^2

Now, rewrite the equation 27^x = 9/3^x as:

(3^3)^x = 3^2 / 3^x

 

Step 2: Simplify Both Sides

Using the power of a power rule (a^m)^n = a^{mn}, simplify the left side:

3^{3x} = {3^2} / {3^x}

Now, use the properties of exponents to simplify the right side. When dividing powers with the same base, subtract the exponents:

3^{3x} = 3^{2-x}

 

Step 3: Equate the Exponents

Since the bases are the same (both are powers of 3), we can equate the exponents:

3x = 2 - x

 

Step 4: Solve for x

Now, solve for x by moving all terms involving x to one side of the equation:

3x + x = 2

4x = 2

Divide both sides by 4 to isolate x:

x = 2/4 = 1/2

 

Step 5: Verify the Solution

It's always important to verify the solution by substituting x= 1/2 back into the original equation.

 Substitute x= ½ into 27^x

27^{1/2} = 3√3

Simplify √27 to 3√3.

Now, substitute x=1/2 into 9/3^x

3^(1/2) /9 = √3/9

 Simplify 3√3 = 3^2 /√3

3√3 = (3*3) /√3

3√3 = 3*(√3*√3)/ √3

3√3 = 3√3

Since both sides of the equation are equal, x=1/2 is the correct solution.

 

Conclusion:

In this equation, 27^x = 9 / 3^x, we used the properties of exponents and simplified both sides to find the solution x =1/2. By expressing everything with the same base, we were able to solve the equation step by step. Mastering these techniques will help you tackle even more complex algebraic equations with confidence.

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