Solve The Equation: 2 W 2 ⋅ 16 W = 4 6 2^{w^2} \cdot 16^w = 4^6 2 W 2 ⋅ 1 6 W = 4 6 W = W = W = Enter Your Answer: □ \square □ Note:- Enter Exact Answers As Integers Or Reduced Fractions.- If There Are Multiple Solutions, Separate Them By Commas.- If No Solution Exists,

Introduction

Exponential equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving the equation $2^{w^2} \cdot 16^w = 4^6$. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Equation

The given equation is $2^{w^2} \cdot 16^w = 4^6$. To simplify this equation, we need to express all the bases in terms of the same base. We know that $16 = 2^4$ and $4 = 2^2$. Substituting these values, we get:

$$2^{w^2} \cdot (2^4)^w = (2^2)^6$$

Using the property of exponents that $(a^m)^n = a^{mn}$, we can simplify the equation further:

$$2^{w^2} \cdot 2^{4w} = 2^{12}$$

Combining Like Terms

Now that we have the same base on both sides of the equation, we can combine the like terms:

$$2^{w^2 + 4w} = 2^{12}$$

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

$$w^2 + 4w = 12$$

Solving the Quadratic Equation

The equation $w^2 + 4w = 12$ is a quadratic equation in the form of $ax^2 + bx + c = 0$. To solve for $w$, we can use the quadratic formula:

$$w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = 4$, and $c = -12$. Plugging these values into the quadratic formula, we get:

$$w = \frac{-4 \pm \sqrt{4^2 - 4(1)(-12)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$w = \frac{-4 \pm \sqrt{16 + 48}}{2}$$

$$w = \frac{-4 \pm \sqrt{64}}{2}$$

$$w = \frac{-4 \pm 8}{2}$$

Finding the Solutions

Now that we have the solutions in terms of the quadratic formula, we can find the values of $w$:

$$w = \frac{-4 + 8}{2} = 2$$

$$w = \frac{-4 - 8}{2} = -6$$

Conclusion

In this article, we solved the exponential equation $2^{w^2} \cdot 16^w = 4^6$ by simplifying it and using the quadratic formula to find the solutions. We found that the solutions are $w = 2$ and $w = -6$. These solutions satisfy the original equation, and we can verify this by plugging them back into the equation.

Final Answer

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Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a power of a variable or a constant. Exponential equations can be written in the form of $a^x = b$, 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 all the bases in terms of the same base. You can do this by using the properties of exponents, such as $(a^m)^n = a^{mn}$ and $a^m \cdot a^n = a^{m+n}$.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula to solve an equation?

A: To use the quadratic formula to solve an equation, you need to identify the coefficients $a$, $b$, and $c$ of the quadratic equation. Then, you can plug these values into the quadratic formula to find the solutions.

Q: What are the steps to solve an exponential equation?

A: The steps to solve an exponential equation are:

1. Simplify the equation by expressing all the bases in terms of the same base.
2. Combine like terms.
3. Equate the exponents.
4. Solve the resulting equation using the quadratic formula.

Q: Can I use the quadratic formula to solve any type of equation?

A: No, the quadratic formula is only used to solve quadratic equations, which are equations of the form $ax^2 + bx + c = 0$. If you have a different type of equation, you may need to use a different method to solve it.

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 properly
• Not combining like terms correctly
• Not equating the exponents correctly
• Not using the quadratic formula correctly

Q: How do I verify the solutions to an exponential equation?

A: To verify the solutions to an exponential equation, you need to plug the solutions back into the original equation and check if they satisfy the equation.

Q: Can I use a calculator to solve exponential equations?

A: Yes, you can use a calculator to solve exponential equations. However, it's always a good idea to check your work by plugging the solutions back into 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 electrical circuits

Conclusion

In this article, we answered some frequently asked questions about solving exponential equations. We covered topics such as simplifying exponential equations, using the quadratic formula, and verifying solutions. We also discussed some common mistakes to avoid and some real-world applications of exponential equations.