Problem 8 - College Algebra

Question

For the following exercises, use like bases to solve the exponential equation. $$ 625 \cdot 5^{3 x+3}=125 $$

Step-by-Step Solution

Answer

The solution for \(x\) is \(-\frac{4}{3}\).

Step 1: Rewrite in terms of like bases

First, we need to express both sides of the equation using the same base. Notice that both 625 and 125 are powers of 5. Specifically, \( 625 = 5^4 \) and \( 125 = 5^3 \). Substitute these values into the equation to get:\[5^4 \cdot 5^{3x+3} = 5^3\]

Step 2: Apply the laws of exponents

Now, apply the laws of exponents. When multiplying like bases, add the exponents:\[5^{4 + (3x + 3)} = 5^3\]Simplify the exponent on the left side:\[5^{3x + 7} = 5^3\]

Step 3: Set the exponents equal

Since the bases are the same, the equation holds true when the exponents are equal. Set the exponents from both sides of the equation as equal:\[3x + 7 = 3\]

Step 4: Solve for x

Subtract 7 from both sides to isolate the term with \(x\):\[3x = 3 - 7\]Simplify:\[3x = -4\]Now, divide both sides by 3 to solve for \(x\):\[x = -\frac{4}{3}\]

Key Concepts

Like Bases

In the world of exponential equations, 'like bases' play a crucial role in simplifying and solving problems. The term refers to expressions that have the same base, meaning the number that is raised to a power, is identical on both sides of the equation. In our exercise, we see this concept in action: the numbers 625 and 125 are both powers of 5.
To rewrite these numbers with the same base, we convert them using powers of 5. For instance,

• 625 can be written as \(5^4\)
• 125 can be written as \(5^3\)

Using like bases simplifies the equation to a standard form, making it easier to apply subsequent mathematical properties, such as the laws of exponents. This approach reduces complexity and allows us to work with just the exponents without distraction from differing base values.

Laws of Exponents

The laws of exponents are foundational rules that help in manipulating and simplifying expressions involving exponents. When dealing with like bases, these rules become highly useful, especially in exponential equations. One essential law used in the exercise is the multiplication of powers property: when you multiply like bases, you simply add their exponents.
Applying this to our example, we see the equation transform:

• \(5^4 \cdot 5^{3x+3}\) becomes \(5^{4 + (3x + 3)}\)
• This simplifies to \(5^{3x + 7}\)

By understanding this law, one can deftly manage the transformation of complex exponential expressions into more manageable forms. These rules are not only vital in solving equations but also in ensuring that mathematical expressions are simplified correctly for further processing.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. When working with exponential equations with like bases, solving becomes straightforward once the bases are identical. The next step involves setting the exponents equal to each other, since if the bases are identical on both sides of the equation, the exponents must be equal too.
Our exercise demonstrates this step-by-step:

• After simplification, we are left with \(5^{3x + 7} = 5^3\)
• Thus, we equate the exponents: \(3x + 7 = 3\)

This step turns the problem into a simple linear equation, which we can solve using basic algebra: isolate the term with the variable, and then find the variable's value by performing operations such as subtraction or division, thereby yielding the correct answer.