Which Expressions Are Equivalent To The One Below? Check All That Apply.$2^5 \cdot 2^x$A. $2^{5x}$B. $2^{5-x}$C. $32 \cdot 2^x$D. $(2 \cdot X)^5$E. $2^{5+x}$F. $4^{5x}$

Introduction

Exponential expressions are a fundamental concept in mathematics, and understanding how to simplify and manipulate them is crucial for solving various mathematical problems. In this article, we will explore the concept of equivalent expressions, specifically focusing on the expression $2^5 \cdot 2^x$. We will examine each of the given options and determine which ones are equivalent to the original expression.

Understanding Exponential Expressions

Before we dive into the equivalent expressions, let's briefly review the concept of exponential expressions. An exponential expression is a mathematical expression that represents a quantity that grows or decays at a constant rate. In the case of the expression $2^5 \cdot 2^x$, we have two exponential terms being multiplied together.

The Original Expression

The original expression is $2^5 \cdot 2^x$. This expression can be simplified using the properties of exponents. Specifically, when multiplying two exponential terms with the same base, we can add their exponents. Therefore, we can simplify the expression as follows:

$$2^5 \cdot 2^x = 2^{5+x}$$

Option A: $2^{5x}$

Let's examine option A, which is $2^{5x}$. This expression is not equivalent to the original expression. The exponent in option A is $5x$, whereas the exponent in the original expression is $5+x$. These two expressions are not equal, and therefore, option A is not a correct equivalent expression.

Option B: $2^{5-x}$

Next, let's examine option B, which is $2^{5-x}$. This expression is also not equivalent to the original expression. The exponent in option B is $5-x$, whereas the exponent in the original expression is $5+x$. These two expressions are not equal, and therefore, option B is not a correct equivalent expression.

Option C: $32 \cdot 2^x$

Now, let's examine option C, which is $32 \cdot 2^x$. This expression can be simplified using the properties of exponents. Specifically, we can rewrite $32$ as $2^5$. Therefore, we can simplify the expression as follows:

$$32 \cdot 2^x = 2^5 \cdot 2^x = 2^{5+x}$$

This expression is equivalent to the original expression, and therefore, option C is a correct equivalent expression.

Option D: $(2 \cdot x)^5$

Next, let's examine option D, which is $(2 \cdot x)^5$. This expression is not equivalent to the original expression. The expression $(2 \cdot x)^5$ represents a quantity that grows or decays at a rate of $5$ times the product of $2$ and $x$. This is not the same as the original expression, which represents a quantity that grows or decays at a rate of $5+x$. Therefore, option D is not a correct equivalent expression.

Option E: $2^{5+x}$

Now, let's examine option E, which is $2^{5+x}$. This expression is equivalent to the original expression. The exponent in option E is $5+x$, which is the same as the exponent in the original expression. Therefore, option E is a correct equivalent expression.

Option F: $4^{5x}$

Finally, let's examine option F, which is $4^{5x}$. This expression is not equivalent to the original expression. The expression $4^{5x}$ represents a quantity that grows or decays at a rate of $5x$ times the base $4$. This is not the same as the original expression, which represents a quantity that grows or decays at a rate of $5+x$. Therefore, option F is not a correct equivalent expression.

Conclusion

In conclusion, the equivalent expressions to the original expression $2^5 \cdot 2^x$ are $2^{5+x}$ and $32 \cdot 2^x$. These two expressions are equivalent to the original expression, and therefore, they are correct equivalent expressions. The other options, $2^{5x}$, $2^{5-x}$, $(2 \cdot x)^5$, and $4^{5x}$, are not equivalent to the original expression, and therefore, they are not correct equivalent expressions.

Key Takeaways

• Exponential expressions can be simplified using the properties of exponents.
• When multiplying two exponential terms with the same base, we can add their exponents.
• The expression $2^5 \cdot 2^x$ can be simplified to $2^{5+x}$.
• The equivalent expressions to the original expression are $2^{5+x}$ and $32 \cdot 2^x$.

Practice Problems

1. Simplify the expression $3^4 \cdot 3^x$.
2. Determine which of the following expressions are equivalent to the original expression $2^5 \cdot 2^x$: $2^{5x}$, $2^{5-x}$, $32 \cdot 2^x$, $(2 \cdot x)^5$, and $4^{5x}$.
3. Simplify the expression $5^3 \cdot 5^y$.

Answer Key

1. $3^{4+x}$
2. $2^{5+x}$ and $32 \cdot 2^x$
3. $5^{3+y}$