Solve $4^ 2x+5}=64$.Answer 1. Simplify And Solve The Equation: $8x^2 + 9x + 2 - \frac{3 {2}x^2$ (Note: The Original Task Contains What Seems To Be A Separate Problem At The End That Isn't Directly Related To Solving The

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 a specific exponential equation, $4^{2x+5}=64$, and provide a step-by-step guide on how to simplify and solve it.

Understanding Exponential Equations

Exponential equations involve variables raised to a power, and the goal is to isolate the variable. In this case, we have an equation with a base of 4 and an exponent of $2x+5$, which is equal to 64. To solve this equation, we need to use the properties of exponents and algebraic manipulations.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by expressing both sides with the same base. We can rewrite 64 as $4^3$, since $4^3 = 64$. Now, our equation becomes:

$$4^{2x+5}=4^3$$

Step 2: Equate the Exponents

Since the bases are the same, we can equate the exponents:

$$2x+5=3$$

Step 3: Solve for x

Now, we need to solve for x by isolating it on one side of the equation. We can do this by subtracting 5 from both sides:

$$2x=-2$$

Step 4: Divide by 2

Finally, we can divide both sides by 2 to solve for x:

$$x=-1$$

Conclusion

In this article, we have solved the exponential equation $4^{2x+5}=64$ by simplifying and equating the exponents. We have shown that the solution to this equation is x = -1. This problem requires a deep understanding of algebraic manipulations and properties of exponents, and it is an essential concept in mathematics.

Additional Tips and Tricks

• When solving exponential equations, it is essential to simplify the equation by expressing both sides with the same base.
• Equating the exponents is a crucial step in solving exponential equations.
• Make sure to isolate the variable on one side of the equation by performing algebraic manipulations.
• Practice solving exponential equations to become proficient in this area of mathematics.

Common Mistakes to Avoid

• Failing to simplify the equation by expressing both sides with the same base.
• Not equating the exponents when the bases are the same.
• Not isolating the variable on one side of the equation by performing algebraic manipulations.

Real-World Applications

Exponential equations have numerous real-world applications, including:

• Modeling population growth and decay
• Calculating compound interest
• Analyzing chemical reactions
• Solving problems in physics and engineering

Conclusion

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable raised to a power, and the goal is to isolate the variable. Exponential equations can be written in the form $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you need to express both sides with the same base. For example, if you have the equation $2^x = 8$, you can rewrite 8 as $2^3$, since $2^3 = 8$. Now, your equation becomes $2^x = 2^3$.

Q: What is the difference between an exponential equation and a linear equation?

A: An exponential equation involves a variable raised to a power, while a linear equation involves a variable multiplied by a coefficient. For example, the equation $2x = 4$ is a linear equation, while the equation $2^x = 4$ is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to follow these steps:

1. Simplify the equation by expressing both sides with the same base.
2. Equate the exponents.
3. Solve for the variable by isolating it on one side of the equation.

Q: What is the property of exponents that allows us to equate the exponents?

A: The property of exponents that allows us to equate the exponents is the fact that if $a^x = a^y$, then $x = y$. This property is known as the "one-to-one" property of exponents.

Q: Can I use logarithms to solve exponential equations?

A: Yes, you can use logarithms to solve exponential equations. In fact, logarithms are a powerful tool for solving 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:

• Failing to simplify the equation by expressing both sides with the same base.
• Not equating the exponents when the bases are the same.
• Not isolating the variable on one side of the equation by performing algebraic manipulations.

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

A: Exponential equations have numerous real-world applications, including:

• Modeling population growth and decay
• Calculating compound interest
• Analyzing chemical reactions
• Solving problems in physics and engineering

Q: Can I use technology to solve exponential equations?

A: Yes, you can use technology to solve exponential equations. Many graphing calculators and computer algebra systems can solve exponential equations and provide solutions.

Conclusion

Solving exponential equations requires a deep understanding of algebraic manipulations and properties of exponents. By following the steps outlined in this article and avoiding common mistakes, you can solve exponential equations with ease. Remember to simplify the equation, equate the exponents, and isolate the variable on one side of the equation. With practice and patience, you will become proficient in solving exponential equations and apply them to real-world problems.