Problem 91 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0.5\).

Step 1: Apply Exponent Addition Rule

Begin by applying the rule \(a^{x} \cdot a^{y} = a^{x+y}\) to combine the terms on the left side of the equation. This gives us the equation: \[5^{2x+4x}=125\]

Step 2: Simplify the Left Side

Next, simplify the left hand side by combining like terms which gives us: \[5^{6x}=125\]

Step 3: Rewrite 125 as base 5

Remember that 125 can be written as \(5^{3}\), so the equation becomes: \[5^{6x}=5^{3}\]

Step 4: Set Exponents Equal to Each Other

Since the equation is in the form \(b^{x}=b^{y}\), we can set the exponents equal to each other. Therefore: \[6x = 3\]

Step 5: Solve for x

Finally, solve for x by dividing both sides by 6. This gives us: \[x = \frac{3}{6} = 0.5\]