Problem 20 - Intermediate Algebra

Question

Solve each equation. $$ 6^{x-2}=36 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\).

Step 1: Recognize the Base

First, notice that the equation is given as \(6^{x-2} = 36\). To solve this for \(x\), recognize that both sides of the equation can be expressed with the same base. The number 36 can be written as \(6^2\).

Step 2: Rewrite the Equation

Substitute \(36\) with its base expression: \(6^{x-2} = 6^2\). Now both sides have the same base.

Step 3: Equate the Exponents

Since the bases are the same, you can set the exponents equal to each other: \(x-2 = 2\).

Step 4: Solve for x

Now solve for \(x\) by adding 2 to both sides of the equation: \(x - 2 + 2 = 2 + 2\), which simplifies to \(x = 4\).

Key Concepts

Equation Solving

Equation solving refers to the process of finding the unknown value that satisfies a given equation. In our example, this unknown value is represented by the variable \( x \). During equation solving, one seeks to isolate the variable on one side of the equation to determine its value.The first step involves observing the equation structure. Identifying similar terms or elements can greatly aid in simplifying the process. Here, the equation \( 6^{x-2} = 36 \) suggests that \( x \) is involved in a power with a base of 6.Remember, the goal is to manipulate the equation so that the variable \( x \) stands alone on one side of the equation. Doing so enables us to determine its exact value and solve the equation.

Exponent Rules

Exponent rules are fundamental when dealing with exponential equations like in the exercise above. These rules dictate how exponents are manipulated and combined, which can make solving the equation much simpler.In our given equation \( 6^{x-2} = 36 \), understanding that 36 is equivalent to \( 6^2 \) is crucial. This rule helps us write the equation with the same base on both sides:

• Product of Powers Rule: When multiplying like bases, add the exponents: \( a^m \cdot a^n = a^{m+n} \).
• Quotient of Powers Rule: When dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).

By recognizing exponent rules, solving exponential equations becomes more straightforward, enabling us to equate the exponents easily, as was done to solve for \( x \).

Algebraic Manipulation

Algebraic manipulation involves rearranging and transforming an equation to isolate the variable and simplify the expression. This skill is essential in solving for unknowns like in the equation \( 6^{x-2} = 6^2 \).Once we have both sides of the equation with the same base, algebraic manipulation focuses on the exponents: by equating \( x-2 = 2 \), we derive an equation that is simple to solve.Let's break down the steps of manipulation:

• Identify similar terms on both sides of the equation. In this case, they are the exponents in \( 6^{x-2} = 6^2 \).
• Apply basic arithmetic operations to isolate the variable. Here, add 2 to both sides of \( x-2 = 2 \) to find \( x \).
• Simplify the result: \( x = 4 \).

Algebraic manipulation allows for transforming and solving equations efficiently by using logical and strategic operations to make the unknown obvious.