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Chapter 12: Problem 91

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

Short Answer

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

Step by step solution

01

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: \[5^{2x+4x}=125\]
02

Simplify the Left Side

Next, simplify the left hand side by combining like terms which gives us: \[5^{6x}=125\]
03

Rewrite 125 as base 5

Remember that 125 can be written as \(5^{3}\), so the equation becomes: \[5^{6x}=5^{3}\]
04

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: \[6x = 3\]
05

Solve for x

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

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