Given The Equation $y = 16 \cdot \left(\frac{1}{4}\right)^n$, Solve For $y$ When $x = 4$.

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the underlying principles. In this article, we will focus on solving a specific exponential equation, $y = 16 \cdot \left(\frac{1}{4}\right)^n$, when $x = 4$. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding Exponential Equations

Exponential equations involve a variable raised to a power, which is often represented by a base and an exponent. In the given equation, $y = 16 \cdot \left(\frac{1}{4}\right)^n$, the base is $\frac{1}{4}$, and the exponent is $n$. The equation states that $y$ is equal to $16$ multiplied by $\left(\frac{1}{4}\right)^n$.

Solving for $y$

To solve for $y$, we need to isolate the variable. In this case, we can start by simplifying the equation. Since $x = 4$, we can substitute $x$ into the equation, but we need to find a way to express $n$ in terms of $x$.

Expressing $n$ in Terms of $x$

We can rewrite the equation as $y = 16 \cdot \left(\frac{1}{4}\right)^x$. This allows us to express $n$ in terms of $x$, where $n = x$.

Substituting $n$ into the Equation

Now that we have expressed $n$ in terms of $x$, we can substitute $n$ into the original equation. This gives us $y = 16 \cdot \left(\frac{1}{4}\right)^x$.

Simplifying the Equation

To simplify the equation, we can use the properties of exponents. Specifically, we can use the fact that $\left(\frac{1}{4}\right)^x = \frac{1}{4^x}$. This allows us to rewrite the equation as $y = 16 \cdot \frac{1}{4^x}$.

Further Simplification

We can further simplify the equation by multiplying $16$ by $\frac{1}{4^x}$. This gives us $y = \frac{16}{4^x}$.

Using the Properties of Exponents

We can use the properties of exponents to simplify the equation further. Specifically, we can use the fact that $\frac{1}{4^x} = 4^{-x}$. This allows us to rewrite the equation as $y = \frac{16}{4^x} = 16 \cdot 4^{-x}$.

Simplifying the Fraction

We can simplify the fraction by dividing $16$ by $4^x$. This gives us $y = 16 \cdot 4^{-x} = 2^{4-x}$.

Final Solution

The final solution is $y = 2^{4-x}$.

Conclusion

In this article, we solved the exponential equation $y = 16 \cdot \left(\frac{1}{4}\right)^n$ when $x = 4$. We broke down the solution into manageable steps, making it easy to follow and understand. We expressed $n$ in terms of $x$, substituted $n$ into the equation, simplified the equation, and used the properties of exponents to arrive at the final solution.

Common Mistakes to Avoid

When solving exponential equations, it's essential to avoid common mistakes. Here are a few to watch out for:

• Not expressing $n$ in terms of $x$: Failing to express $n$ in terms of $x$ can lead to incorrect solutions.
• Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.
• Not using the properties of exponents: Failing to use the properties of exponents can lead to incorrect solutions.

Real-World Applications

Exponential equations have numerous real-world applications. Here are a few examples:

• Population growth: Exponential equations can be used to model population growth, where the population grows at a rate proportional to the current population.
• Financial modeling: Exponential equations can be used to model financial growth, where the growth rate is proportional to the current value.
• Science and engineering: Exponential equations can be used to model a wide range of phenomena, from the growth of bacteria to the decay of radioactive materials.

Final Thoughts

Q: What is an exponential equation?

A: An exponential equation is a mathematical equation that involves a variable raised to a power, often represented by a base and an exponent. In the equation $y = 16 \cdot \left(\frac{1}{4}\right)^n$, the base is $\frac{1}{4}$, and the exponent is $n$.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable. In this case, we can start by simplifying the equation. Since $x = 4$, we can substitute $x$ into the equation, but we need to find a way to express $n$ in terms of $x$.

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 constant. For example, the equation $y = 2x$ is a linear equation, while the equation $y = 2^x$ is an exponential equation.

Q: Can I use the same methods to solve exponential equations as I would for linear equations?

A: No, you cannot use the same methods to solve exponential equations as you would for linear equations. Exponential equations require a different set of techniques and properties, such as the properties of exponents.

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

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

• Not expressing $n$ in terms of $x$
• Not simplifying the equation
• Not using the properties of exponents

Q: How do I use the properties of exponents to solve exponential equations?

A: To use the properties of exponents to solve exponential equations, you need to understand the rules of exponents, such as:

• $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
• $\left(a^m\right)^n = a^{mn}$
• $a^m \cdot a^n = a^{m+n}$

Q: Can I use exponential equations to model real-world phenomena?

A: Yes, exponential equations can be used to model a wide range of real-world phenomena, such as population growth, financial growth, and the decay of radioactive materials.

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

A: To apply exponential equations to real-world problems, you need to:

• Identify the variables and constants in the equation
• Use the properties of exponents to simplify the equation
• Use the equation to model the real-world phenomenon
• Solve for the variable

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

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

• Population growth: $y = 2^x$
• Financial growth: $y = 1.05^x$
• Decay of radioactive materials: $y = 0.5^x$

Q: How do I choose the correct method to solve an exponential equation?

A: To choose the correct method to solve an exponential equation, you need to:

• Identify the type of equation (e.g. linear, quadratic, exponential)
• Use the properties of exponents to simplify the equation
• Use the equation to model the real-world phenomenon
• Solve for the variable

Q: Can I use technology to solve exponential equations?

A: Yes, you can use technology to solve exponential equations. Many calculators and computer software programs have built-in functions for solving exponential equations.

Q: How do I use technology to solve exponential equations?

A: To use technology to solve exponential equations, you need to:

• Enter the equation into the calculator or computer software program
• Use the built-in functions to simplify the equation
• Use the equation to model the real-world phenomenon
• Solve for the variable