Problem 93 - College Algebra Essentials

Question

Solve each equation. $$5^{2 x} \cdot 5^{4 x}=125$$

Step-by-Step Solution

Answer

\(\)The solution for \(x\) is 0.5.

Step 1: Rewrite the Equation in a Simplified Form

The given equation is \(5^{2x} \cdot 5^{4x} = 125\). Using the property of exponents, \(a^m \cdot a^n = a^{m+n}\), we can rewrite the equation as \(5^{(2x+4x)} = 125\). This simplifies to \(5^{6x} = 125\).

Step 2: Express the Numbers with the Same Base

Now, write 125 in terms of base 5. 125 is the same as \(5^3\). This allows us to write the equation as \(5^{6x} = 5^3\).

Step 3: Equate the Exponents

Once we have the same base on both sides of the equation, we can set the exponents equal to each other. So, \(6x = 3\).

Step 4: Solving for \(x\)

Finally, solve for \(x\) by dividing both sides of the equation by 6. This gives us \(x = 3/6 = 0.5\).