Applying the Product of Powers Property

Oct 13, 2025 by recoveryshake.com 17 views

Introduction

When it comes to simplifying expressions involving exponents, it's essential to understand the rules of exponentiation and how to apply them to solve problems. In this article, we'll focus on simplifying the expression $5^3 \times 5^{-5}$ using the properties of exponents.

Understanding Exponents

Before we dive into the solution, let's quickly review the basics of exponents. An exponent is a small number that is placed above and to the right of a number, indicating how many times the base number should be multiplied by itself. For example, in the expression $5^3$ , the exponent $3$ indicates that the base number $5$ should be multiplied by itself three times: $5^3 = 5 \times 5 \times 5 = 125$ .

Applying the Product of Powers Property

To simplify the expression $5^3 \times 5^{-5}$ , we can use the product of powers property, which states that when multiplying two numbers with the same base, we can add their exponents. In this case, both numbers have the base $5$ , so we can add their exponents: $5^3 \times 5^{-5} = 5^{3+(-5)}$ .

Simplifying the Exponent

Now that we have added the exponents, we can simplify the resulting expression. When adding a positive and a negative exponent, we subtract the absolute value of the negative exponent from the positive exponent: $5^{3+(-5)} = 5^{3-5} = 5^{-2}$ .

Understanding Negative Exponents

A negative exponent indicates that we should take the reciprocal of the base number. In this case, the base number is $5$ , so we should take the reciprocal of $5$ to the power of $2$ : $5^{-2} = \frac{1}{5^2}$ .

Conclusion

In conclusion, to simplify the expression $5^3 \times 5^{-5}$ , we applied the product of powers property to add the exponents, resulting in $5^{3+(-5)} = 5^{-2}$ . We then simplified the negative exponent by taking the reciprocal of the base number, resulting in $\frac{1}{5^2}$ . Therefore, the correct answer is:

The Correct Answer

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

Discussion

This problem requires a good understanding of the properties of exponents, including the product of powers property and the concept of negative exponents. By applying these properties, we can simplify complex expressions and arrive at the correct solution.

Example Use Cases

This problem is a great example of how to apply the product of powers property to simplify expressions involving exponents. In real-world applications, this concept is used in a variety of fields, including physics, engineering, and computer science.

Tips and Tricks

When simplifying expressions involving exponents, it's essential to remember the following tips and tricks:

Always apply the product of powers property when multiplying two numbers with the same base.
When adding a positive and a negative exponent, subtract the absolute value of the negative exponent from the positive exponent.
Negative exponents indicate that we should take the reciprocal of the base number.

By following these tips and tricks, you can simplify complex expressions involving exponents and arrive at the correct solution.

Common Mistakes

When simplifying expressions involving exponents, it's easy to make mistakes. Here are some common mistakes to avoid:

Failing to apply the product of powers property when multiplying two numbers with the same base.
Adding the exponents instead of subtracting the absolute value of the negative exponent.
Not taking the reciprocal of the base number when dealing with negative exponents.

By avoiding these common mistakes, you can ensure that your solutions are accurate and correct.

Conclusion

In conclusion, simplifying the expression $5^3 \times 5^{-5}$ requires a good understanding of the properties of exponents, including the product of powers property and the concept of negative exponents. By applying these properties, we can simplify complex expressions and arrive at the correct solution. Remember to always apply the product of powers property, subtract the absolute value of the negative exponent, and take the reciprocal of the base number when dealing with negative exponents. With practice and patience, you can become proficient in simplifying expressions involving exponents.

Introduction

In our previous article, we explored how to simplify the expression $5^3 \times 5^{-5}$ using the properties of exponents. In this article, we'll take a Q&A approach to provide additional insights and clarify any doubts you may have.

Q: What is the product of powers property?

A: The product of powers property states that when multiplying two numbers with the same base, we can add their exponents. In other words, if we have two numbers in the form $a^m$ and $a^n$ , we can simplify the product as $a^{m+n}$ .

Q: How do I apply the product of powers property to simplify the expression $5^3 \times 5^{-5}$ ?

A: To apply the product of powers property, we simply add the exponents: $5^3 \times 5^{-5} = 5^{3+(-5)} = 5^{-2}$ .

Q: What does a negative exponent mean?

A: A negative exponent indicates that we should take the reciprocal of the base number. In the case of $5^{-2}$ , we should take the reciprocal of $5$ to the power of $2$ , resulting in $\frac{1}{5^2}$ .

Q: Can I simplify the expression $5^3 \times 5^{-5}$ by multiplying the numbers instead of using the product of powers property?

A: While you can multiply the numbers, it's not the most efficient way to simplify the expression. By using the product of powers property, you can avoid multiplying large numbers and arrive at the correct solution more quickly.

Q: What if I have an expression with multiple bases, such as $2^3 \times 3^4$ ? How do I simplify it?

A: When dealing with multiple bases, you can apply the product of powers property separately for each base. In the case of $2^3 \times 3^4$ , you can simplify the expression as $2^3 \times 3^4 = (2^3) \times (3^4) = 8 \times 81 = 648$ .

Q: Can I use the product of powers property to simplify expressions with fractions, such as $\frac{2^3}{3^4}$ ?

A: Yes, you can use the product of powers property to simplify expressions with fractions. In the case of $\frac{2^3}{3^4}$ , you can simplify the expression as $\frac{2^3}{3^4} = \frac{8}{3^4} = \frac{8}{81}$ .

Q: What if I have an expression with a negative base, such as $-2^3$ ? How do I simplify it?

A: When dealing with a negative base, you can simplify the expression by applying the exponent to the base. In the case of $-2^3$ , you can simplify the expression as $-2^3 = -(2^3) = -8$ .

Q: Can I use the product of powers property to simplify expressions with variables, such as $x^3 \times y^4$ ?

A: Yes, you can use the product of powers property to simplify expressions with variables. In the case of $x^3 \times y^4$ , you can simplify the expression as $x^3 \times y^4 = (x^3) \times (y^4) = x^3y^4$ .