Solve For The Variables Shown In The Steps By Simplifying The Expression Below:$\[ \frac{2^5 \cdot 8^4}{16} = \frac{2^5 \cdot \left(2^a\right)^4}{2^4} = \frac{2^5 \cdot 2^b}{2^4} = 2^c. \\]Fill In The Blanks For The Variables:- \[$a =

**Solving for Variables in Algebraic Expressions: A Step-by-Step Guide** ===========================================================

Introduction

Algebraic expressions are a fundamental concept in mathematics, and solving for variables is a crucial skill that students and professionals need to master. In this article, we will explore the steps involved in solving for variables in algebraic expressions, using a specific example to illustrate the process.

The Problem

The problem we will be solving is:

{ \frac{2^5 \cdot 8^4}{16} = \frac{2^5 \cdot \left(2^a\right)^4}{2^4} = \frac{2^5 \cdot 2^b}{2^4} = 2^c. \}

Our goal is to fill in the blanks for the variables $a$, $b$, and $c$.

Step 1: Simplify the Expression

To solve for the variables, we need to simplify the expression. We can start by rewriting the expression using the properties of exponents.

{ \frac{2^5 \cdot 8^4}{16} = \frac{2^5 \cdot \left(2^3\right)^4}{2^4} = \frac{2^5 \cdot 2^{12}}{2^4} = 2^{5+12-4} = 2^{13}. \}

Step 2: Identify the Value of $a$

Now that we have simplified the expression, we can identify the value of $a$. We know that $8^4 = \left(2^3\right)^4$, so we can rewrite the expression as:

{ \frac{2^5 \cdot \left(2^3\right)^4}{2^4} = 2^{5+12-4} = 2^{13}. \}

From this, we can see that $a = 3$, since $\left(2^3\right)^4 = 2^{12}$.

Step 3: Identify the Value of $b$

Next, we need to identify the value of $b$. We know that $2^5 \cdot 2^b = 2^{13}$, so we can rewrite the expression as:

{ 2^5 \cdot 2^b = 2^{13}. \}

From this, we can see that $b = 8$, since $2^5 \cdot 2^8 = 2^{13}$.

Step 4: Identify the Value of $c$

Finally, we need to identify the value of $c$. We know that $2^c = 2^{13}$, so we can rewrite the expression as:

{ 2^c = 2^{13}. \}

From this, we can see that $c = 13$, since $2^c = 2^{13}$.

Conclusion

In this article, we have walked through the steps involved in solving for variables in algebraic expressions. We have used a specific example to illustrate the process, and have identified the values of $a$, $b$, and $c$. By following these steps, you can solve for variables in algebraic expressions and become more confident in your math skills.

Frequently Asked Questions

Q: What is the value of $a$ in the expression $\frac{2^5 \cdot 8^4}{16}$?

A: The value of $a$ is 3, since $\left(2^3\right)^4 = 2^{12}$.

Q: What is the value of $b$ in the expression $\frac{2^5 \cdot 8^4}{16}$?

A: The value of $b$ is 8, since $2^5 \cdot 2^8 = 2^{13}$.

Q: What is the value of $c$ in the expression $\frac{2^5 \cdot 8^4}{16}$?

A: The value of $c$ is 13, since $2^c = 2^{13}$.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you can use the properties of exponents, such as the product rule and the power rule.

Q: How do I solve for variables in an algebraic expression?

A: To solve for variables in an algebraic expression, you can follow the steps outlined in this article, including simplifying the expression and identifying the values of the variables.

Q: What are some common mistakes to avoid when solving for variables in algebraic expressions?

A: Some common mistakes to avoid when solving for variables in algebraic expressions include:

• Not simplifying the expression before solving for variables
• Not using the properties of exponents correctly
• Not identifying the values of the variables correctly

By following these steps and avoiding common mistakes, you can become more confident in your math skills and solve for variables in algebraic expressions with ease.