Problem 11 - Intermediate Algebra

Question

For Problems \(1-34\), solve each equation. $$ 6^{3 x-1}=36 $$

Step-by-Step Solution

Answer

The solution is \(x = 1\).

Step 1: Express 36 as a power of 6

Since 36 is 6 squared, we can express it as a power of 6: \[ 36 = 6^2 \] This will help in comparing exponents of the same base.

Step 2: Equate the exponents

Given that the equation is \(6^{3x-1} = 36\) and we've rewritten 36 as \(6^2\), we can now equate the exponents of the base since the bases are the same.\[ 3x-1 = 2 \]

Step 3: Solve for x

To isolate \(x\), add 1 to both sides of the equation:\[ 3x - 1 + 1 = 2 + 1 \]Which simplifies to:\[ 3x = 3 \]Next, divide both sides by 3 to solve for \(x\):\[ x = 1 \]

Key Concepts

Expressing Numbers as Powers

Expressing numbers as powers is a useful skill in mathematics, especially when dealing with exponential equations. Here, our task is to express numbers as a power of another number—usually the base of the equation we're working with. In many cases, simplifying an equation into the same base allows us to compare or manipulate exponents with greater ease.

To express a number as a power, determine the base and see how many times it must be multiplied by itself to get the number in question. For example, in our exercise, we identified that 36 is a power of 6, specifically,

• 36 = 6 x 6
• Which translates to 36 = 6^2

This simplification aligns the expression 36 with the base of the original exponential equation (6), allowing a straightforward path to the next steps.

Solving for Variables

Solving for variables means isolating the variable on one side of the equation, allowing us to calculate its value accurately. After expressing numbers as powers of a common base, as seen in exponential equations, the next step usually involves solving for the variable present in the exponent.

Continuing with our exercise, after aligning our bases, we compared and made use of the equation

• 3x - 1 = 2

To solve for the variable, follow these steps:

• Add or subtract constants to both sides of the equation. Here, we added 1 to both sides: 3x - 1 + 1 = 2 + 1, which simplifies to 3x = 3.
• Divide both sides by the coefficient of x. So, dividing both sides by 3 results in x = 1.

This step-by-step simplification is the cornerstone of solving for variables effectively.

Equating Exponents

Equating exponents is a technique used when two exponential expressions with like bases are set equal to one another. Once you have the exponents in terms of the same base, you can simplify and solve the equation by merely equating their exponents.

For our example, once 36 was rewritten as a power of 6, we had the expression 6^(3x-1) = 6^2.
By having the same base on both sides of an equation, we can focus on the exponents:

• 3x - 1 = 2

This allows us to disregard the base and directly solve for the variable within the exponent. It's a straightforward technique that removes complexities, narrowing the problem down to a simple linear equation.