Simplify The Expression $5^3 \times 5^{-5}$.A. $\frac{1}{5}$ B. $5^2$ C. $-5^2$ D. $\frac{1}{5^2}$

Understanding Exponents and Negative Exponents

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this case, we're given the expression $5^3 \times 5^{-5}$, and we need to simplify it. To do this, we'll use the rule for multiplying numbers with the same base, which states that when we multiply two numbers with the same base, we add their exponents.

The Rule for Multiplying Numbers with the Same Base

The rule for multiplying numbers with the same base is as follows:

$a^m \times a^n = a^{m+n}$

where $a$ is the base, and $m$ and $n$ are the exponents.

Applying the Rule to the Given Expression

Now, let's apply this rule to the given expression $5^3 \times 5^{-5}$. We can see that both numbers have the same base, which is $5$. Therefore, we can add their exponents:

$5^3 \times 5^{-5} = 5^{3+(-5)}$

Simplifying the Exponent

When we add $3$ and $-5$, we get $-2$. Therefore, the expression simplifies to:

$5^{3+(-5)} = 5^{-2}$

Understanding Negative Exponents

A negative exponent is a fraction with the base in the denominator. In other words, $a^{-n} = \frac{1}{a^n}$. Therefore, we can rewrite $5^{-2}$ as:

$5^{-2} = \frac{1}{5^2}$

Simplifying the Expression

Now that we've simplified the expression, we can see that it matches one of the answer choices. Therefore, the correct answer is:

$\frac{1}{5^2}$

Conclusion

In this article, we simplified the expression $5^3 \times 5^{-5}$ using the rule for multiplying numbers with the same base. We added the exponents and then simplified the resulting expression using the rule for negative exponents. The correct answer is $\frac{1}{5^2}$.

Frequently Asked Questions

• What is the rule for multiplying numbers with the same base? The rule for multiplying numbers with the same base is $a^m \times a^n = a^{m+n}$.
• How do we simplify an expression with a negative exponent? We can rewrite a negative exponent as a fraction with the base in the denominator, i.e., $a^{-n} = \frac{1}{a^n}$.
• What is the correct answer to the given expression? The correct answer is $\frac{1}{5^2}$.

Final Answer

The final answer is $\boxed{\frac{1}{5^2}}$.

Understanding Exponents and Negative Exponents

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this case, we're given the expression $5^3 \times 5^{-5}$, and we need to simplify it. To do this, we'll use the rule for multiplying numbers with the same base, which states that when we multiply two numbers with the same base, we add their exponents.

Q&A: Simplifying Exponents

Q: What is the rule for multiplying numbers with the same base?

A: The rule for multiplying numbers with the same base is $a^m \times a^n = a^{m+n}$.

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

A: We can rewrite a negative exponent as a fraction with the base in the denominator, i.e., $a^{-n} = \frac{1}{a^n}$.

Q: What is the correct answer to the given expression?

A: The correct answer is $\frac{1}{5^2}$.

Q: Can you explain the concept of negative exponents in more detail?

A: A negative exponent is a fraction with the base in the denominator. For example, $a^{-n} = \frac{1}{a^n}$. This means that when we have a negative exponent, we can rewrite it as a fraction with the base in the denominator.

Q: How do we apply the rule for multiplying numbers with the same base to the given expression?

A: To apply the rule, we add the exponents of the two numbers with the same base. In this case, we have $5^3 \times 5^{-5}$, so we add the exponents: $5^{3+(-5)} = 5^{-2}$.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{1}{5^2}$.

Common Mistakes to Avoid

• Not understanding the rule for multiplying numbers with the same base
• Not recognizing the concept of negative exponents
• Not applying the rule correctly to the given expression

Tips for Simplifying Exponents

• Make sure to understand the rule for multiplying numbers with the same base
• Recognize the concept of negative exponents and how to rewrite them as fractions
• Apply the rule correctly to the given expression

Conclusion

In this article, we simplified the expression $5^3 \times 5^{-5}$ using the rule for multiplying numbers with the same base. We added the exponents and then simplified the resulting expression using the rule for negative exponents. We also answered common questions and provided tips for simplifying exponents.

Frequently Asked Questions

• What is the rule for multiplying numbers with the same base? The rule for multiplying numbers with the same base is $a^m \times a^n = a^{m+n}$.
• How do we simplify an expression with a negative exponent? We can rewrite a negative exponent as a fraction with the base in the denominator, i.e., $a^{-n} = \frac{1}{a^n}$.
• What is the correct answer to the given expression? The correct answer is $\frac{1}{5^2}$.

Final Answer

The final answer is $\boxed{\frac{1}{5^2}}$.