Solve The Equation:$121^{x+2}=\frac{1}{11^{3x+1}}$

Feb 26, 2025 by ADMIN 51 views

**Solving Exponential Equations: A Step-by-Step Guide**

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and properties of exponents. In this article, we will focus on solving the equation $121^{x+2}=\frac{1}{11^{3x+1}}$ , which involves both positive and negative exponents. We will break down the solution into manageable steps, using a combination of algebraic manipulations and properties of exponents.

Understanding the Equation

The given equation is $121^{x+2}=\frac{1}{11^{3x+1}}$ . To begin solving this equation, we need to understand the properties of exponents and how to manipulate them. The base of the exponent on the left-hand side is 121, which can be expressed as $11^2$ . Similarly, the base of the exponent on the right-hand side is 11.

121^{x+2} = (11^2)^{x+2}

Using the property of exponents that $(a^b)^c = a^{bc}$ , we can simplify the left-hand side of the equation:

(11^2)^{x+2} = 11^{2(x+2)}

Now, we can rewrite the equation as:

11^{2(x+2)} = \frac{1}{11^{3x+1}}

Simplifying the Equation

To simplify the equation further, we can use the property of exponents that $\frac{1}{a^b} = a^{-b}$ . Applying this property to the right-hand side of the equation, we get:

11^{2(x+2)} = 11^{-(3x+1)}

Now, we can equate the exponents on both sides of the equation:

2(x+2) = -(3x+1)

Solving for x

To solve for x, we can start by simplifying the equation:

2x + 4 = -3x - 1

Next, we can add 3x to both sides of the equation to get:

5x + 4 = -1

Now, we can subtract 4 from both sides of the equation to get:

5x = -5

Finally, we can divide both sides of the equation by 5 to solve for x:

x = -1

Conclusion

In this article, we have solved the equation $121^{x+2}=\frac{1}{11^{3x+1}}$ using a combination of algebraic manipulations and properties of exponents. We have shown that the solution to this equation is x = -1. This example illustrates the importance of understanding the properties of exponents and how to manipulate them to solve complex equations.

Properties of Exponents

Exponents are a fundamental concept in mathematics, and understanding their properties is crucial for solving complex equations. Some of the key properties of exponents include:

Product of Powers: $a^b \cdot a^c = a^{b+c}$
Power of a Power: $(a^b)^c = a^{bc}$
Quotient of Powers: $\frac{a^b}{a^c} = a^{b-c}$
Negative Exponent: $a^{-b} = \frac{1}{a^b}$

Real-World Applications

Exponential equations have numerous real-world applications, including:

Finance: Exponential equations are used to model population growth, compound interest, and inflation.
Science: Exponential equations are used to model chemical reactions, population growth, and the spread of diseases.
Engineering: Exponential equations are used to model the behavior of electrical circuits, mechanical systems, and thermal systems.

Final Thoughts

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponent, which is a number or expression raised to a power. Exponential equations can be written in the form $a^b = c$ , where $a$ is the base, $b$ is the exponent, and $c$ is the result.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable (usually $x$ ) by using algebraic manipulations and properties of exponents. This may involve simplifying the equation, using the product of powers property, or using the power of a power property.

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

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is raised to a negative power. For example, $a^b$ is a positive exponent, while $a^{-b}$ is a negative exponent.

Q: How do I simplify an exponential equation with a negative exponent?

A: To simplify an exponential equation with a negative exponent, you can use the property of exponents that $a^{-b} = \frac{1}{a^b}$ . This allows you to rewrite the equation with a positive exponent.

Q: Can I use logarithms to solve exponential equations?

A: Yes, logarithms can be used to solve exponential equations. By taking the logarithm of both sides of the equation, you can eliminate the exponent and solve for the variable.

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

A: Some common mistakes to avoid when solving exponential equations include:

Not simplifying the equation: Make sure to simplify the equation before solving for the variable.
Not using the correct properties of exponents: Use the correct properties of exponents, such as the product of powers property and the power of a power property.
Not checking the solution: Make sure to check the solution to ensure that it satisfies the original equation.

Q: How do I apply exponential equations to real-world problems?

A: Exponential equations can be applied to a wide range of real-world problems, including:

Finance: Exponential equations can be used to model population growth, compound interest, and inflation.
Science: Exponential equations can be used to model chemical reactions, population growth, and the spread of diseases.
Engineering: Exponential equations can be used to model the behavior of electrical circuits, mechanical systems, and thermal systems.

Q: What are some examples of exponential equations in real-world problems?

A: Some examples of exponential equations in real-world problems include:

Population growth: The population of a city grows exponentially, with a growth rate of 2% per year.
Compound interest: An investment earns an interest rate of 5% per year, compounded annually.
Inflation: The price of a product increases exponentially, with an inflation rate of 3% per year.

Q: How do I choose the correct method for solving an exponential equation?

A: To choose the correct method for solving an exponential equation, you need to consider the following factors:

The form of the equation: If the equation is in the form $a^b = c$ , you can use the properties of exponents to simplify the equation.
The presence of logarithms: If the equation involves logarithms, you can use logarithmic properties to simplify the equation.
The complexity of the equation: If the equation is complex, you may need to use a combination of methods to solve it.

Conclusion

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and properties of exponents. By following the steps outlined in this article, you can solve complex exponential equations and apply them to real-world problems. Remember to always use the properties of exponents to simplify the equation and solve for the variable. With practice and patience, you can become proficient in solving exponential equations and apply them to a wide range of fields.