Select All The Expressions That Are Equivalent To $5^3 \cdot 5^{-4}$.A. $\frac{1}{5^{-12}}$ B. $\frac{1}{5^{-1}}$ C. $5^{-1}$ D. $\frac{1}{5}$

Introduction

Exponential expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $5^3 \cdot 5^{-4}$ and identifying equivalent forms.

Understanding Exponents

Before we dive into simplifying the expression, let's briefly review the concept of exponents. An exponent is a small number that is written above and to the right of a base number. It represents the number of times the base number is multiplied by itself. For example, $5^3$ means $5$ multiplied by itself $3$ times, which is equal to $5 \cdot 5 \cdot 5 = 125$.

Simplifying the Expression

Now, let's simplify the expression $5^3 \cdot 5^{-4}$. To do this, we can use the rule of exponents that states when multiplying two numbers with the same base, we add their exponents. In this case, the base is $5$, and the exponents are $3$ and $-4$. Therefore, we can simplify the expression as follows:

$$5^3 \cdot 5^{-4} = 5^{3 + (-4)} = 5^{-1}$$

Equivalent Forms

Now that we have simplified the expression to $5^{-1}$, let's explore equivalent forms. An equivalent form is an expression that has the same value as the original expression. In this case, we can rewrite $5^{-1}$ in several equivalent forms:

• Fractional form: We can rewrite $5^{-1}$ as a fraction by using the rule that $a^{-n} = \frac{1}{a^n}$. Therefore, $5^{-1} = \frac{1}{5^1} = \frac{1}{5}$.
• Negative exponent form: We can also rewrite $5^{-1}$ using a negative exponent. Since $a^{-n} = \frac{1}{a^n}$, we can rewrite $5^{-1}$ as $\frac{1}{5^1} = \frac{1}{5}$.
• Inverse form: Finally, we can rewrite $5^{-1}$ as an inverse form by using the rule that $a^{-n} = \frac{1}{a^n}$. Therefore, $5^{-1} = \frac{1}{5^1} = \frac{1}{5}$.

Conclusion

In conclusion, we have simplified the expression $5^3 \cdot 5^{-4}$ to $5^{-1}$ and explored equivalent forms. We have shown that $5^{-1}$ can be rewritten in several equivalent forms, including fractional form, negative exponent form, and inverse form. These equivalent forms are useful for solving various mathematical problems and can help us to simplify complex expressions.

Answer

Based on our analysis, we can conclude that the correct answer is:

• D. $\frac{1}{5}$

This is because $\frac{1}{5}$ is an equivalent form of $5^{-1}$, which is the simplified expression of $5^3 \cdot 5^{-4}$.

Discussion

Now that we have simplified the expression and identified equivalent forms, let's discuss some related topics.

• Exponent rules: We have used several exponent rules to simplify the expression, including the rule that $a^{-n} = \frac{1}{a^n}$ and the rule that $a^m \cdot a^n = a^{m+n}$. These rules are essential for simplifying exponential expressions and are used extensively in mathematics.
• Equivalent forms: We have shown that $5^{-1}$ can be rewritten in several equivalent forms, including fractional form, negative exponent form, and inverse form. These equivalent forms are useful for solving various mathematical problems and can help us to simplify complex expressions.
• Simplifying expressions: We have simplified the expression $5^3 \cdot 5^{-4}$ to $5^{-1}$ using exponent rules. This process involves applying the rules of exponents to simplify the expression and identify equivalent forms.

Practice Problems

Now that we have simplified the expression and identified equivalent forms, let's practice solving some related problems.

• Problem 1: Simplify the expression $2^4 \cdot 2^{-3}$.
• Problem 2: Rewrite the expression $3^{-2}$ in fractional form.
• Problem 3: Simplify the expression $4^2 \cdot 4^{-1}$.

Solution

• Problem 1: Using the rule that $a^m \cdot a^n = a^{m+n}$, we can simplify the expression as follows: $2^4 \cdot 2^{-3} = 2^{4 + (-3)} = 2^1 = 2$.
• Problem 2: Using the rule that $a^{-n} = \frac{1}{a^n}$, we can rewrite the expression as follows: $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.
• Problem 3: Using the rule that $a^m \cdot a^n = a^{m+n}$, we can simplify the expression as follows: $4^2 \cdot 4^{-1} = 4^{2 + (-1)} = 4^1 = 4$.

Conclusion

Q: What is the rule for simplifying exponential expressions?

A: The rule for simplifying exponential expressions is to use the rule that $a^m \cdot a^n = a^{m+n}$ and the rule that $a^{-n} = \frac{1}{a^n}$.

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

A: To simplify an expression with a negative exponent, you can use the rule that $a^{-n} = \frac{1}{a^n}$. For example, $5^{-1} = \frac{1}{5^1} = \frac{1}{5}$.

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

A: A positive exponent means that the base number is being multiplied by itself a certain number of times, while a negative exponent means that the base number is being divided by itself a certain number of times.

Q: How do I rewrite an expression with a negative exponent in fractional form?

A: To rewrite an expression with a negative exponent in fractional form, you can use the rule that $a^{-n} = \frac{1}{a^n}$. For example, $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

Q: What is the inverse form of an expression with a negative exponent?

A: The inverse form of an expression with a negative exponent is the same as the fractional form. For example, $5^{-1} = \frac{1}{5}$.

Q: Can I simplify an expression with a negative exponent by using the rule that $a^m \cdot a^n = a^{m+n}$?

A: No, you cannot simplify an expression with a negative exponent by using the rule that $a^m \cdot a^n = a^{m+n}$. This rule only applies to expressions with positive exponents.

Q: How do I simplify an expression with multiple negative exponents?

A: To simplify an expression with multiple negative exponents, you can use the rule that $a^{-n} = \frac{1}{a^n}$ and the rule that $a^m \cdot a^n = a^{m+n}$. For example, $5^{-3} \cdot 5^{-2} = 5^{(-3) + (-2)} = 5^{-5} = \frac{1}{5^5} = \frac{1}{3125}$.

Q: Can I simplify an expression with a negative exponent by using the rule that $a^m \cdot a^n = a^{m+n}$ and then taking the reciprocal of the result?

A: Yes, you can simplify an expression with a negative exponent by using the rule that $a^m \cdot a^n = a^{m+n}$ and then taking the reciprocal of the result. For example, $5^{-3} \cdot 5^{-2} = 5^{(-3) + (-2)} = 5^{-5} = \frac{1}{5^5} = \frac{1}{3125}$.

Q: How do I know when to use the rule that $a^{-n} = \frac{1}{a^n}$ and when to use the rule that $a^m \cdot a^n = a^{m+n}$?

A: You should use the rule that $a^{-n} = \frac{1}{a^n}$ when you have an expression with a negative exponent, and you should use the rule that $a^m \cdot a^n = a^{m+n}$ when you have an expression with multiple positive exponents.

Q: Can I simplify an expression with a negative exponent by using the rule that $a^m \cdot a^n = a^{m+n}$ and then taking the reciprocal of the result, even if the expression has multiple negative exponents?

A: Yes, you can simplify an expression with multiple negative exponents by using the rule that $a^m \cdot a^n = a^{m+n}$ and then taking the reciprocal of the result. For example, $5^{-3} \cdot 5^{-2} = 5^{(-3) + (-2)} = 5^{-5} = \frac{1}{5^5} = \frac{1}{3125}$.

Conclusion

In conclusion, we have answered some common questions about simplifying exponential expressions. We have shown that the rule for simplifying exponential expressions is to use the rule that $a^m \cdot a^n = a^{m+n}$ and the rule that $a^{-n} = \frac{1}{a^n}$. We have also shown that you can simplify an expression with a negative exponent by using the rule that $a^{-n} = \frac{1}{a^n}$ and that you can simplify an expression with multiple negative exponents by using the rule that $a^m \cdot a^n = a^{m+n}$ and then taking the reciprocal of the result.