Solve The Equation:$ (\sqrt{2})^{x+5} = 4^x $The Solution Set Is $ { \square } $(Type An Integer Or A Fraction.)

$$

Introduction

In this article, we will delve into solving the equation $(\sqrt{2})^{x+5} = 4^x$. This equation involves exponents and radicals, and we will use various mathematical techniques to simplify and solve it. The solution set will be presented in the form of a set, where the value of $x$ will be an integer or a fraction.

Understanding the Equation

The given equation is $(\sqrt{2})^{x+5} = 4^x$. To begin solving this equation, we need to understand the properties of exponents and radicals. The expression $(\sqrt{2})^{x+5}$ can be rewritten as $2^{(x+5)/2}$, using the property of radicals that $\sqrt{a} = a^{1/2}$. Similarly, the expression $4^x$ can be rewritten as $2^{2x}$, using the property of exponents that $(a^m)^n = a^{mn}$.

Simplifying the Equation

Now that we have rewritten the equation in terms of exponents, we can simplify it further. We have:

$$2^{(x+5)/2} = 2^{2x}$$

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

$$\frac{x+5}{2} = 2x$$

Solving for $x$

To solve for $x$, we can start by multiplying both sides of the equation by 2 to eliminate the fraction:

$$x+5 = 4x$$

Next, we can subtract $x$ from both sides to get:

$$5 = 3x$$

Finally, we can divide both sides by 3 to solve for $x$:

$$x = \frac{5}{3}$$

Conclusion

In this article, we have solved the equation $(\sqrt{2})^{x+5} = 4^x$ using various mathematical techniques. We have rewritten the equation in terms of exponents, simplified it, and solved for $x$. The solution set is $\left\{\frac{5}{3}\right\}$.

Final Answer

The final answer is $\boxed{\frac{5}{3}}$.

Discussion

The equation $(\sqrt{2})^{x+5} = 4^x$ is a classic example of an exponential equation. It involves exponents and radicals, and requires careful manipulation to solve. The solution set is a single value, which is an integer or a fraction. This type of equation is commonly encountered in algebra and mathematics, and requires a deep understanding of mathematical concepts and techniques.

Tips and Tricks

• When solving exponential equations, it is essential to rewrite the equation in terms of exponents.
• Use the properties of exponents and radicals to simplify the equation.
• Equate the exponents when the bases are the same.
• Solve for the variable by isolating it on one side of the equation.

Related Topics

• Exponential equations
• Radical equations
• Algebra
• Mathematics

References

• [1] "Algebra" by Michael Artin
• [2] "Mathematics" by Richard Courant
• [3] "Exponential Equations" by Wolfram MathWorld

Further Reading

• For more information on exponential equations, see the Wolfram MathWorld article on "Exponential Equations".
• For more information on radical equations, see the Wolfram MathWorld article on "Radical Equations".
• For more information on algebra, see the book "Algebra" by Michael Artin.
• For more information on mathematics, see the book "Mathematics" by Richard Courant.
  

$$

Introduction

In our previous article, we solved the equation $(\sqrt{2})^{x+5} = 4^x$ using various mathematical techniques. In this article, we will provide a Q&A section to address any questions or concerns that readers may have.

Q&A

Q: What is the solution set for the equation $(\sqrt{2})^{x+5} = 4^x$?

A: The solution set for the equation $(\sqrt{2})^{x+5} = 4^x$ is $\left\{\frac{5}{3}\right\}$.

Q: How do I rewrite the equation in terms of exponents?

A: To rewrite the equation in terms of exponents, you can use the property of radicals that $\sqrt{a} = a^{1/2}$. For example, the expression $(\sqrt{2})^{x+5}$ can be rewritten as $2^{(x+5)/2}$.

Q: What is the property of exponents that allows me to equate the exponents when the bases are the same?

A: The property of exponents that allows you to equate the exponents when the bases are the same is $(a^m)^n = a^{mn}$. This property states that when you raise a power to a power, you can multiply the exponents.

Q: How do I solve for $x$ in the equation $\frac{x+5}{2} = 2x$?

A: To solve for $x$ in the equation $\frac{x+5}{2} = 2x$, you can start by multiplying both sides of the equation by 2 to eliminate the fraction. This gives you $x+5 = 4x$. Next, you can subtract $x$ from both sides to get $5 = 3x$. Finally, you can divide both sides by 3 to solve for $x$.

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

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

• Not rewriting the equation in terms of exponents
• Not equating the exponents when the bases are the same
• Not solving for the variable by isolating it on one side of the equation
• Not checking the solution to make sure it satisfies the original equation

Q: What are some real-world applications of exponential equations?

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

• Modeling population growth
• Modeling chemical reactions
• Modeling financial investments
• Modeling physical phenomena such as radioactive decay

Conclusion

In this article, we have provided a Q&A section to address any questions or concerns that readers may have about solving the equation $(\sqrt{2})^{x+5} = 4^x$. We have covered topics such as rewriting the equation in terms of exponents, equating the exponents when the bases are the same, and solving for the variable. We have also discussed common mistakes to avoid and real-world applications of exponential equations.

Final Answer

The final answer is $\boxed{\frac{5}{3}}$.

Discussion

The equation $(\sqrt{2})^{x+5} = 4^x$ is a classic example of an exponential equation. It involves exponents and radicals, and requires careful manipulation to solve. The solution set is a single value, which is an integer or a fraction. This type of equation is commonly encountered in algebra and mathematics, and requires a deep understanding of mathematical concepts and techniques.

Tips and Tricks

• When solving exponential equations, it is essential to rewrite the equation in terms of exponents.
• Use the properties of exponents and radicals to simplify the equation.
• Equate the exponents when the bases are the same.
• Solve for the variable by isolating it on one side of the equation.

Related Topics

• Exponential equations
• Radical equations
• Algebra
• Mathematics

References

• [1] "Algebra" by Michael Artin
• [2] "Mathematics" by Richard Courant
• [3] "Exponential Equations" by Wolfram MathWorld

Further Reading

• For more information on exponential equations, see the Wolfram MathWorld article on "Exponential Equations".
• For more information on radical equations, see the Wolfram MathWorld article on "Radical Equations".
• For more information on algebra, see the book "Algebra" by Michael Artin.
• For more information on mathematics, see the book "Mathematics" by Richard Courant.