Solve For \[$x\$\] In The Equation:$\[ 25^x = \frac{1}{125} \\]

Introduction

In mathematics, solving equations involving exponents is a crucial skill that can be applied to various real-world problems. One such problem is solving for the variable $x$ in the equation $25^x = \frac{1}{125}$. This equation involves a base of 25 and an exponent of $x$, and the goal is to isolate the variable $x$ and find its value. In this article, we will explore the steps to solve for $x$ in this equation and provide a clear understanding of the mathematical concepts involved.

Understanding the Equation

The given equation is $25^x = \frac{1}{125}$. To begin solving this equation, we need to understand the properties of exponents and how to manipulate them. The base of the exponent is 25, and the exponent is $x$. The right-hand side of the equation is $\frac{1}{125}$, which can be rewritten as $5^{-3}$.

Rewriting the Equation

We can rewrite the equation as $25^x = 5^{-3}$. This step involves expressing the fraction $\frac{1}{125}$ in terms of a power of 5, which is a common base for exponents. By rewriting the equation in this form, we can use the properties of exponents to simplify the equation and solve for $x$.

Using Exponent Properties

To solve for $x$, we can use the property of exponents that states $a^m = a^n$ if and only if $m = n$. In this case, we have $25^x = 5^{-3}$. Since the bases are different, we cannot directly compare the exponents. However, we can rewrite the base 25 as a power of 5, which is $5^2$. This allows us to rewrite the equation as $(5^2)^x = 5^{-3}$.

Simplifying the Equation

Using the property of exponents that states $(a^m)^n = a^{mn}$, we can simplify the left-hand side of the equation as $5^{2x} = 5^{-3}$. This step involves applying the exponent rule to the base 5, which allows us to combine the exponents.

Equating Exponents

Since the bases are the same, we can equate the exponents of the two sides of the equation. This gives us the equation $2x = -3$. To solve for $x$, we can divide both sides of the equation by 2, which gives us $x = -\frac{3}{2}$.

Conclusion

In this article, we have solved for the variable $x$ in the equation $25^x = \frac{1}{125}$. We began by rewriting the equation in terms of a common base, using the properties of exponents to simplify the equation, and equating the exponents of the two sides. The final solution is $x = -\frac{3}{2}$, which represents the value of the variable $x$ that satisfies the given equation.

Applications of Exponent Rules

Exponent rules are essential in mathematics and have numerous applications in various fields, including physics, engineering, and computer science. Some of the key applications of exponent rules include:

• Solving equations: Exponent rules are used to solve equations involving exponents, such as the equation $25^x = \frac{1}{125}$.
• Graphing functions: Exponent rules are used to graph functions involving exponents, such as the function $f(x) = 2^x$.
• Modeling real-world problems: Exponent rules are used to model real-world problems involving exponential growth or decay, such as population growth or radioactive decay.

Common Mistakes to Avoid

When solving equations involving exponents, there are several common mistakes to avoid:

• Not rewriting the equation in terms of a common base: Failing to rewrite the equation in terms of a common base can make it difficult to apply exponent rules and solve the equation.
• Not using exponent properties correctly: Failing to use exponent properties correctly can lead to incorrect solutions or incorrect conclusions.
• Not checking the solution: Failing to check the solution can lead to incorrect conclusions or incorrect applications of the solution.

Final Thoughts

Solving equations involving exponents is a crucial skill that can be applied to various real-world problems. By understanding the properties of exponents and how to manipulate them, we can solve equations involving exponents and apply the solutions to real-world problems. In this article, we have solved for the variable $x$ in the equation $25^x = \frac{1}{125}$ and provided a clear understanding of the mathematical concepts involved.

Introduction

In our previous article, we solved for the variable $x$ in the equation $25^x = \frac{1}{125}$. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have about solving equations involving exponents.

Q: What is the difference between a base and an exponent?

A: In an exponential expression, the base is the number being raised to a power, and the exponent is the power to which the base is being raised. For example, in the expression $2^3$, the base is 2 and the exponent is 3.

Q: How do I rewrite an equation in terms of a common base?

A: To rewrite an equation in terms of a common base, you need to express both sides of the equation in terms of the same base. For example, if you have the equation $25^x = \frac{1}{125}$, you can rewrite it as $5^{2x} = 5^{-3}$ by expressing both sides in terms of the base 5.

Q: What is the property of exponents that states $a^m = a^n$ if and only if $m = n$?

A: This property is known as the one-to-one property of exponents. It states that if two exponential expressions have the same base and the same value, then their exponents must be equal.

Q: How do I simplify an exponential expression using the property $(a^m)^n = a^{mn}$?

A: To simplify an exponential expression using this property, you need to multiply the exponents together. For example, if you have the expression $(2^3)^4$, you can simplify it to $2^{12}$ by multiplying the exponents together.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is being raised to a power, while a negative exponent indicates that the base is being taken to a power. For example, in the expression $2^3$, the exponent is positive, while in the expression $2^{-3}$, the exponent is negative.

Q: How do I solve an equation involving a negative exponent?

A: To solve an equation involving a negative exponent, you need to rewrite the equation in terms of a positive exponent. For example, if you have the equation $2^{-x} = 4$, you can rewrite it as $2^x = \frac{1}{4}$ by taking the reciprocal of both sides.

Q: What are some common mistakes to avoid when solving equations involving exponents?

A: Some common mistakes to avoid when solving equations involving exponents include:

• Not rewriting the equation in terms of a common base
• Not using exponent properties correctly
• Not checking the solution

Q: How do I apply exponent rules to real-world problems?

A: Exponent rules can be applied to real-world problems involving exponential growth or decay. For example, if you are modeling the growth of a population, you can use exponent rules to calculate the population size at a given time.

Q: What are some examples of real-world problems that involve exponent rules?

A: Some examples of real-world problems that involve exponent rules include:

• Modeling population growth or decay
• Calculating the amount of money in a savings account with compound interest
• Determining the amount of radiation emitted by a radioactive substance

Conclusion

In this Q&A article, we have provided answers to common questions about solving equations involving exponents. We hope that this article has helped to clarify any doubts or questions that readers may have had about exponent rules and their applications.