Which Expressions Are Equivalent To The One Below? Check All That Apply. 25 X 5 X \frac{25^x}{5^x} 5 X 2 5 X ​ A. 5 X 5^x 5 X B. ( 25 5 ) X \left(\frac{25}{5}\right)^x ( 5 25 ​ ) X C. 5 X ⋅ 5 X 5 X \frac{5^x \cdot 5^x}{5^x} 5 X 5 X ⋅ 5 X ​ D. 25 X 25^x 2 5 X E. 5 F. ( 25 − 5 ) X (25-5)^x ( 25 − 5 ) X

Introduction

In mathematics, simplifying exponential expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. When dealing with exponential expressions, it's essential to identify equivalent expressions that can be simplified to a more manageable form. In this article, we will explore the concept of equivalent expressions and simplify the given expression $\frac{25^x}{5^x}$ to identify the correct equivalent expressions.

Understanding Exponential Expressions

Exponential expressions are mathematical expressions that involve a base raised to a power. The base is the number that is being raised to the power, and the exponent is the number that is being raised to the power. For example, in the expression $2^3$, the base is 2 and the exponent is 3.

Simplifying the Given Expression

The given expression is $\frac{25^x}{5^x}$. To simplify this expression, we can use the properties of exponents. Specifically, we can use the property that states $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is the base and $m$ and $n$ are the exponents.

Using this property, we can rewrite the given expression as:

$$\frac{25^x}{5^x} = \frac{(5^2)^x}{5^x} = \frac{5^{2x}}{5^x} = 5^{2x-x} = 5^x$$

Identifying Equivalent Expressions

Now that we have simplified the given expression, we can identify the equivalent expressions. The equivalent expressions are the expressions that have the same value as the simplified expression.

Let's examine each of the options:

A. $5^x$ - This is the simplified expression that we obtained earlier.

B. $\left(\frac{25}{5}\right)^x$ - This expression can be rewritten as $\left(5\right)^x$, which is equivalent to $5^x$.

C. $\frac{5^x \cdot 5^x}{5^x}$ - This expression can be simplified by canceling out the common factor of $5^x$, leaving us with $5^x$.

D. $25^x$ - This expression can be rewritten as $(5^2)^x$, which is equivalent to $5^{2x}$. However, we know that $5^{2x} = 5^x \cdot 5^x$, so this expression is not equivalent to $5^x$.

E. 5 - This expression is not equivalent to $5^x$ because it does not have the variable $x$.

F. $(25-5)^x$ - This expression can be rewritten as $20^x$, which is not equivalent to $5^x$.

Conclusion

In conclusion, the equivalent expressions to the given expression $\frac{25^x}{5^x}$ are:

• A. $5^x$
• B. $\left(\frac{25}{5}\right)^x$
• C. $\frac{5^x \cdot 5^x}{5^x}$

These expressions have the same value as the simplified expression and can be used as equivalent expressions in mathematical problems.

Final Thoughts

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is a mathematical expression that involves a base raised to a power, whereas a polynomial expression is a mathematical expression that involves a sum of terms, each of which is a constant or a variable raised to a non-negative integer power.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you can use the properties of exponents, such as the product rule, the quotient rule, and the power rule. You can also use the fact that $a^m \cdot a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, and $(a^m)^n = a^{mn}$.

Q: What is the product rule for exponents?

A: The product rule for exponents states that $a^m \cdot a^n = a^{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents.

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is the base and $m$ and $n$ are the exponents.

Q: What is the power rule for exponents?

A: The power rule for exponents states that $(a^m)^n = a^{mn}$, where $a$ is the base and $m$ and $n$ are the exponents.

Q: How do I simplify an expression with multiple bases?

A: To simplify an expression with multiple bases, you can use the fact that $a^m \cdot b^n = (ab)^{m+n}$, where $a$ and $b$ are the bases and $m$ and $n$ are the exponents.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base is being raised to a power, whereas a negative exponent indicates that the base is being taken to a power.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the fact that $a^{-n} = \frac{1}{a^n}$, where $a$ is the base and $n$ is the exponent.

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

A: An exponential expression is a mathematical expression that involves a base raised to a power, whereas a logarithmic expression is a mathematical expression that involves the inverse operation of exponentiation.

Q: How do I simplify an expression with a logarithm?

A: To simplify an expression with a logarithm, you can use the fact that $\log_a{b} = c$ is equivalent to $a^c = b$, where $a$ is the base and $b$ is the argument.

Q: What are some common mistakes to avoid when simplifying exponential expressions?

A: Some common mistakes to avoid when simplifying exponential expressions include:

• Forgetting to use the properties of exponents
• Not simplifying the expression completely
• Making errors when applying the product rule, quotient rule, or power rule
• Not checking the domain of the expression

Conclusion

In conclusion, simplifying exponential expressions is an essential skill in mathematics that helps us solve complex problems and understand the underlying concepts. By using the properties of exponents and identifying equivalent expressions, we can simplify complex expressions and make them more manageable. We hope that this article has provided you with a comprehensive guide to simplifying exponential expressions and has helped you to avoid common mistakes.