Problem 5 - Intermediate Algebra

Question

Fill in the blanks. $$ \text { If } 6^{4 x}=6^{-2}, \text { then } 4 x= $$

Step-by-Step Solution

Answer

4x = -2

Step 1: Recognize the Exponents

Since both sides of the equation \( 6^{4x} = 6^{-2} \) have the same base, you can equate the exponents. This leads us to the equation \( 4x = -2 \).

Step 2: Solve for x

To find the value of \( x \), we need to solve the equation \( 4x = -2 \). Divide both sides of the equation by 4 to isolate \( x \).\[ x = \frac{-2}{4} \]\[ x = -\frac{1}{2} \]

Step 3: Write Solution for 4x

Since \( x = -\frac{1}{2} \), substitute this value back into \( 4x \) to find the answer.\[ 4x = 4 \times -\frac{1}{2} \]\[ 4x = -2 \]

Key Concepts

Exponents

Exponents can be thought of as a way to express repeated multiplication. For example, the term \( 3^4 \) means multiplying 3 by itself four times: \( 3 \times 3 \times 3 \times 3 \). In this way, exponents provide a shorthand method to express large numbers or repeated multiplications.

• **Base:** The big number, like the 3 in \( 3^4 \), which is multiplied by itself the number of times indicated by the exponent.
• **Exponent:** The small number above the base that tells us how many times to multiply.

When dealing with algebraic equations involving exponents, certain rules make calculations more manageable. One important rule is that if two expressions with the same base are equal (like \( a^m = a^n \)), then their exponents must be equal too, leading to the equation \( m = n \). This rule was used in the exercise when we equated the exponents \( 4x = -2 \) since both sides had the base 6.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. The process typically involves using arithmetic operations such as addition, subtraction, multiplication, or division to both sides of the equation until the variable is isolated.To solve the equation from the exercise:

• Start with \( 4x = -2 \).
• The goal is to solve for \( x \) by isolating it.
• Since \( x \) is multiplied by 4, do the opposite operation; divide both sides by 4 to get \( x = \frac{-2}{4} \).
• Simplify the expression to find \( x = -\frac{1}{2} \).

Remember, the key is to perform the same operation on both sides, keeping the equation balanced.

Isolate Variable

Isolating the variable is a critical step in solving equations. The aim is to get the variable alone on one side of the equation, making it easier to understand and solve. This often means performing a series of reversals of the operations done on the variable.In the equation \( 4x = -2 \), where \( x \) must be isolated:

• Identify what is being done to \( x \). Here, it is being multiplied by 4.
• To reverse this operation, divide both sides of the equation by 4.
• When this is done, you achieve: \( x = \frac{-2}{4} \).
• Simplify the fraction to \( x = -\frac{1}{2} \) for the final result.

Using these steps, you can see that isolating a variable often involves undoing addition, subtraction, multiplication, or division. The strategy is to "do the opposite" and work backwards from the operation applied to the variable.