In Exercises 93–102, 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 on the left side: $5^{\mathrm{2x}}\cdot 5^{\mathrm{4x}}=125$.
2. Use the property of exponents that states ${}^{}m\cdot {}^{}n={}^{}m+n$ to combine the left side: $5^{2x\; +\; 4x}=125$.
3. Simplify the exponent sum: $2x\; +\; 4x\; =\; 6x$, so the equation becomes $5^{6x}=125$.
4. Express 125 as a power of 5, since 125 is a power of 5: $125\; =3^{}$.
5. Set the exponents equal to each other because the bases are the same: $6x\; =\; 3$, then solve for $x$ by dividing both sides by 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, a^m * a^n = a^(m+n). Understanding these properties is essential to simplify and solve exponential equations.

Expressing Numbers with a Common Base

To solve exponential equations, it helps to rewrite all terms with the same base. For instance, 125 can be expressed as 5^3 since 125 = 5 × 5 × 5. This allows the equation to be set up with equal bases, making it easier to compare exponents.

Solving Linear Equations

After rewriting the equation with a common base and applying exponent rules, the problem reduces to solving a linear equation in terms of the variable exponent. This involves isolating the variable and performing algebraic operations to find its value.