Question

Solve each equation. $$ 5^{x+3}=\frac{1}{5} $$

Short answer

x = -4

Step-by-step solution

- Understand the equation

The given equation is \( 5^{x+3} = \frac{1}{5} \). We need to find the value of \( x \) that satisfies this equation.

- Express the equation using a common base

Recognize that \( \frac{1}{5} \) can be written as \( 5^{-1} \). Thus, the equation becomes \( 5^{x+3} = 5^{-1} \).

- Equate the exponents

Since the bases on both sides of the equation are the same, we can set the exponents equal to each other: \( x + 3 = -1 \).

- Solve for \( x \)

Subtract 3 from both sides of the equation to isolate \( x \): \( x = -1 - 3 \). Therefore, \( x = -4 \).

Key concepts

headline of the respective core concept

Solving equations is a fundamental skill in mathematics that involves finding the value of the variable that makes the equation true. In our exercise, we are given the equation \( 5^{x+3} = \frac{1}{5} \). To solve this, we need to determine the value of \( x \).

• First, we need to express both sides of the equation using the same base.
• Once we have a common base, we can then set the exponents equal to each other.
• Finally, we solve for the variable to find the solution.

This process involves several important steps. Let's explore these concepts more deeply in the following sections.

headline of the respective core concept

A common base is essential in solving exponential equations. In our given problem, we notice that \( \frac{1}{5} \) is an exponent of 5 in a negative form. Therefore, we can rewrite our equation from \( 5^{x+3} = \frac{1}{5} \) to \( 5^{x+3} = 5^{-1} \).

• This step allows us to compare the exponents directly.
• Finding a common base simplifies the process.
• Working with common bases transforms the equation into a more manageable form.

Simplifying an equation using common bases is not only logical but also makes the steps that follow more intuitive. Understanding how to convert and manipulate exponents helps solve these equations effectively.

headline of the respective core concept

Equating exponents becomes straightforward once the bases are the same. With our equation rewritten as \( 5^{x+3} = 5^{-1} \), we can now set the exponents equal to each other, because if \( a^m = a^n \), then \( m = n \). Therefore, we write:
\( x + 3 = -1 \)

• This step eliminates the exponential base and brings us to a linear equation.
• We solve for \( x \) by isolating it on one side of the equation.
• In this case, subtract 3 from both sides to obtain: \( x = -1 - 3 \).

Thus, \( x = -4 \). Equating and solving exponents reduces the complexity of exponential equations, leading us to a clear solution.