Problem 23 - Intermediate Algebra

Question

Solve each equation. $$ 5^{4 x}=\frac{1}{125} $$

Step-by-Step Solution

Answer

\(x = -\frac{3}{4}\)

Step 1: Convert to Exponential Form

First, notice that the equation is in the form of an exponential equation, where the base on the left side is 5. The number on the right side, \(\frac{1}{125}\), can be rewritten using the same base. Since 125 is equal to \(5^3\), \(\frac{1}{125}\) can be written as \(5^{-3}\). This transforms the equation into: \[ 5^{4x} = 5^{-3} \]

Step 2: Equate the Exponents

The bases on both sides of the equation are the same (base 5). This allows us to set the exponents equal to each other. Thus, we can write:\[ 4x = -3 \]

Step 3: Solve for x

To find the value of \(x\), divide both sides of the equation by 4:\[ x = \frac{-3}{4} \]

Key Concepts

Equation Solving

Solving equations is a crucial step in understanding mathematics, especially when dealing with algebra. An equation is simply a statement indicating that two expressions are equal, usually including an unknown variable that we need to find. In the given problem, the equation involves equal expressions: one side with a variable in the exponent and the other side as a number. The goal is to find the value of the variable that makes the two sides equal. To solve such an equation, it's essential to manipulate the expressions correctly:

• Identify the variable: This helps focus the solution process on isolating this variable.
• Use operations to simplify: Addition, subtraction, multiplication, and division help in breaking down more complex parts into simpler ones.
• Check your solution: After finding potential values, substitute back into the original equation to ensure both sides match.

This problem shows how to set the exponents equal by using the property of exponential equations, leading to a simpler algebraic form to solve.

Exponents

Exponents are compact ways to represent repeated multiplication. In mathematics, they help express larger numbers efficiently and simplify calculations. An exponent tells us how many times a number, known as the base, is multiplied by itself. For example, in the term \(5^x\), 5 is the base, and \(x\) is the exponent. Understanding exponents involves familiarizing yourself with key rules:

• Multiplication: \(a^m \times a^n = a^{m+n}\)
• Division: \(a^m \div a^n = a^{m-n}\)
• Power of a power: \((a^m)^n = a^{m \cdot n}\)

In this task, recognizing \(\frac{1}{125}\) as \(5^{-3}\) allows transforming the equation into a simpler form without changing its truth, paving the way for solving the equation by matching exponents.

Algebra

Algebra is the branch of mathematics dealing with symbols and rules for manipulating those symbols. It forms the basis for most mathematical calculations and problem-solving techniques. Within algebra, solving equations is fundamental. To master algebra, one should:

• Understand variables: Represent unknowns and constants in equations.
• Apply operations wisely: Use addition, subtraction, multiplication, and division to manipulate expressions and equations.
• Recognize equivalent expressions: Equations may look complex but often can be simplified to a form that is easier to solve.

In our exercise, the algebraic manipulation of the equation \(5^{4x} = \frac{1}{125}\) involved rewriting as \(5^{4x} = 5^{-3}\) so the basis matched, and therefore, the exponents could be equated. This step-by-step reasoning is crucial in handling equations in algebra efficiently and accurately.