Solve The Following Equation:${ \left(\frac{1}{4}\right)^{x+2} = (16)^{4x+1} }$

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Introduction

Solving equations involving exponents can be a challenging task, especially when dealing with different bases and exponents. In this article, we will focus on solving the equation $\left(\frac{1}{4}\right)^{x+2} = (16)^{4x+1}$, which involves both fractional and integer exponents. We will use various mathematical techniques, including exponent properties and logarithms, to simplify the equation and find the solution.

Understanding the Equation

The given equation is $\left(\frac{1}{4}\right)^{x+2} = (16)^{4x+1}$. To begin solving this equation, we need to understand the properties of exponents and how they can be manipulated. The equation involves two different bases: $\frac{1}{4}$ and $16$. We can rewrite $16$ as $4^2$, which will help us simplify the equation.

Simplifying the Equation

We can start by rewriting the equation using the fact that $16 = 4^2$. This gives us:

$$\left(\frac{1}{4}\right)^{x+2} = (4^2)^{4x+1}$$

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

$$(4^2)^{4x+1} = 4^{2(4x+1)}$$

Now, we can rewrite the equation as:

$$\left(\frac{1}{4}\right)^{x+2} = 4^{8x+2}$$

Using Exponent Properties

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

$$\frac{1^{x+2}}{4^{x+2}} = 4^{8x+2}$$

Since $1$ raised to any power is still $1$, we can simplify the left-hand side of the equation:

$$\frac{1}{4^{x+2}} = 4^{8x+2}$$

Using Logarithms

To solve for $x$, we can use logarithms to eliminate the exponents. Taking the logarithm of both sides of the equation, we get:

$$\log\left(\frac{1}{4^{x+2}}\right) = \log(4^{8x+2})$$

Using the property of logarithms that $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$, we can simplify the left-hand side of the equation:

$$\log(1) - \log(4^{x+2}) = \log(4^{8x+2})$$

Since $\log(1) = 0$, we can simplify the equation further:

$$-\log(4^{x+2}) = \log(4^{8x+2})$$

Solving for x

To solve for $x$, we can use the property of logarithms that $\log(a^b) = b\log(a)$. Applying this property to both sides of the equation, we get:

$$-(x+2)\log(4) = (8x+2)\log(4)$$

Since $\log(4)$ is a constant, we can divide both sides of the equation by $\log(4)$:

$$-(x+2) = 8x+2$$

Now, we can solve for $x$:

$$-x-2 = 8x+2$$

$$-9x = 4$$

$$x = -\frac{4}{9}$$

Conclusion

In this article, we solved the equation $\left(\frac{1}{4}\right)^{x+2} = (16)^{4x+1}$ using various mathematical techniques, including exponent properties and logarithms. We simplified the equation by rewriting the bases and using exponent properties, and then used logarithms to eliminate the exponents. Finally, we solved for $x$ by isolating the variable. The solution to the equation is $x = -\frac{4}{9}$.

Additional Tips and Tricks

• When dealing with equations involving exponents, it's often helpful to rewrite the bases using exponent properties.
• Logarithms can be a powerful tool for solving equations involving exponents.
• Be careful when simplifying equations involving exponents, as small mistakes can lead to incorrect solutions.

Frequently Asked Questions

• Q: What is the solution to the equation $\left(\frac{1}{4}\right)^{x+2} = (16)^{4x+1}$? A: The solution to the equation is $x = -\frac{4}{9}$.
• Q: How do I simplify equations involving exponents? A: You can simplify equations involving exponents by rewriting the bases using exponent properties and using logarithms to eliminate the exponents.
• 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 failing to rewrite the bases using exponent properties and making errors when simplifying the equation.
  

Introduction

Solving equations involving exponents can be a challenging task, especially when dealing with different bases and exponents. In this article, we will answer some frequently asked questions about solving equations involving exponents, including tips and tricks for simplifying the equation and finding the solution.

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

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

Q: How do I simplify equations involving exponents?

A: You can simplify equations involving exponents by rewriting the bases using exponent properties and using logarithms to eliminate the exponents. For example, if you have the equation $\left(\frac{1}{4}\right)^{x+2} = (16)^{4x+1}$, you can rewrite the bases as $4^{-(x+2)} = 4^{2(4x+1)}$.

Q: What is the property of exponents that states $(a^b)^c = a^{bc}$?

A: This property is called the power of a power property. It states that when you raise a power to another power, you can multiply the exponents. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

Q: How do I use logarithms to solve equations involving exponents?

A: You can use logarithms to solve equations involving exponents by taking the logarithm of both sides of the equation. This will allow you to eliminate the exponents and solve for the variable. For example, if you have the equation $2^x = 8$, you can take the logarithm of both sides to get $x = \log_2(8)$.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, the equation $x = \log_2(8)$ is a logarithmic equation, while the equation $2^x = 8$ is an exponential equation.

Q: How do I solve a logarithmic equation?

A: You can solve a logarithmic equation by using the definition of a logarithm. For example, if you have the equation $x = \log_2(8)$, you can rewrite it as $2^x = 8$ and solve for $x$.

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 failing to rewrite the bases using exponent properties, making errors when simplifying the equation, and forgetting to check the domain of the logarithm.

Q: How do I check the domain of a logarithm?

A: You can check the domain of a logarithm by making sure that the argument of the logarithm is positive. For example, if you have the equation $x = \log_2(8)$, you can check the domain by making sure that $8$ is positive.

Q: What is the difference between a one-to-one function and a many-to-one function?

A: A one-to-one function is a function that maps each input to a unique output, while a many-to-one function is a function that maps multiple inputs to the same output. For example, the function $f(x) = x^2$ is a many-to-one function, while the function $f(x) = x$ is a one-to-one function.

Q: How do I determine if a function is one-to-one or many-to-one?

A: You can determine if a function is one-to-one or many-to-one by checking if it passes the horizontal line test. If the function passes the horizontal line test, it is one-to-one. If it does not pass the horizontal line test, it is many-to-one.

Q: What is the significance of one-to-one functions in mathematics?

A: One-to-one functions are significant in mathematics because they have a unique inverse. This means that if you have a one-to-one function, you can find its inverse by swapping the x and y values.

Q: How do I find the inverse of a one-to-one function?

A: You can find the inverse of a one-to-one function by swapping the x and y values. For example, if you have the function $f(x) = x^2$, you can find its inverse by swapping the x and y values to get $f^{-1}(x) = \sqrt{x}$.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two functions that are related to each other. The inverse of a function is a function that undoes the action of the original function. For example, if you have the function $f(x) = x^2$, its inverse is $f^{-1}(x) = \sqrt{x}$.

Q: How do I determine if a function is invertible?

A: You can determine if a function is invertible by checking if it is one-to-one. If the function is one-to-one, it is invertible. If it is not one-to-one, it is not invertible.

Q: What is the significance of invertible functions in mathematics?

A: Invertible functions are significant in mathematics because they have a unique inverse. This means that if you have an invertible function, you can find its inverse by swapping the x and y values.

Q: How do I find the inverse of an invertible function?

A: You can find the inverse of an invertible function by swapping the x and y values. For example, if you have the function $f(x) = x^2$, you can find its inverse by swapping the x and y values to get $f^{-1}(x) = \sqrt{x}$.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two functions that are related to each other. The inverse of a function is a function that undoes the action of the original function. For example, if you have the function $f(x) = x^2$, its inverse is $f^{-1}(x) = \sqrt{x}$.

Q: How do I determine if a function is invertible?

A: You can determine if a function is invertible by checking if it is one-to-one. If the function is one-to-one, it is invertible. If it is not one-to-one, it is not invertible.

Q: What is the significance of invertible functions in mathematics?

A: Invertible functions are significant in mathematics because they have a unique inverse. This means that if you have an invertible function, you can find its inverse by swapping the x and y values.

Q: How do I find the inverse of an invertible function?

A: You can find the inverse of an invertible function by swapping the x and y values. For example, if you have the function $f(x) = x^2$, you can find its inverse by swapping the x and y values to get $f^{-1}(x) = \sqrt{x}$.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two functions that are related to each other. The inverse of a function is a function that undoes the action of the original function. For example, if you have the function $f(x) = x^2$, its inverse is $f^{-1}(x) = \sqrt{x}$.

Q: How do I determine if a function is invertible?

A: You can determine if a function is invertible by checking if it is one-to-one. If the function is one-to-one, it is invertible. If it is not one-to-one, it is not invertible.

Q: What is the significance of invertible functions in mathematics?

A: Invertible functions are significant in mathematics because they have a unique inverse. This means that if you have an invertible function, you can find its inverse by swapping the x and y values.

Q: How do I find the inverse of an invertible function?

A: You can find the inverse of an invertible function by swapping the x and y values. For example, if you have the function $f(x) = x^2$, you can find its inverse by swapping the x and y values to get $f^{-1}(x) = \sqrt{x}$.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two functions that are related to each other. The inverse of a function is a function that undoes the action of the original function. For example, if you have the function $f(x) = x^2$, its inverse is $f^{-1}(x) = \sqrt{x}$.

Q: How do I determine if a function is invertible?

A: You can determine if a function is invertible by checking if it is one-to-one. If the function is one-to-one, it is invertible. If it is not one-to-one, it is not invertible.

Q: What is the significance of invertible functions in mathematics?

A: Invertible functions are significant in mathematics because they have a unique inverse. This means that if you have an invertible function, you can find its inverse by swapping the x and y values.

Q: How do I find the inverse of an invertible function?

A