Solve each equation. 5^2x ⋅ 5^4x=125

Verified step by step guidance

1. Recognize that the equation involves exponential expressions with the same base. The equation is \(5^{2x} \cdot 5^{4x} = 125\).
2. Use the property of exponents that states when multiplying like bases, you add the exponents: \(a^m \cdot a^n = a^{m+n}\). So rewrite the left side as \(5^{2x + 4x} = 5^{6x}\).
3. Rewrite the right side, 125, as a power of 5. Since \(125 = 5^3\), the equation becomes \(5^{6x} = 5^3\).
4. Since the bases are the same and the expressions are equal, set the exponents equal to each other: \(6x = 3\).
5. Solve the equation \(6x = 3\) for \(x\) by dividing both sides by 6: \(x = \frac{3}{6}\).

Key Concepts

Properties of Exponents

This concept involves rules for manipulating expressions with exponents, such as multiplying powers with the same base by adding their exponents. For example, 5^(2x) * 5^(4x) equals 5^(2x + 4x) = 5^(6x). Understanding these properties simplifies solving exponential equations.

Expressing Numbers with the Same Base

To solve exponential equations, it helps to rewrite all terms with the same base. Since 125 can be written as 5^3, the equation 5^(6x) = 125 becomes 5^(6x) = 5^3, allowing us to set the exponents equal to each other for solving.

Solving Linear Equations

After equating the exponents, the problem reduces to solving a linear equation like 6x = 3. This involves isolating the variable by performing inverse operations, such as dividing both sides by 6, to find the value of x.