Use the properties of exponents and roots

Oct 13, 2025 by magicalceylontours.com 1 views

Understanding the Problem

When dealing with exponents and roots, it's essential to understand the properties and rules that govern them. In this problem, we're given the expression $\sqrt[5]{13^3}$ and asked to find an equivalent expression from the given options.

Properties of Exponents and Roots

To solve this problem, we need to recall the properties of exponents and roots. Specifically, we need to remember that:

$\sqrt[n]{a} = a^{\frac{1}{n}}$
$a^m \cdot a^n = a^{m+n}$
$(a^m)^n = a^{m \cdot n}$

Applying the Properties

Using the properties mentioned above, we can rewrite the given expression $\sqrt[5]{13^3}$ as follows:

$\sqrt[5]{13^3} = (13^3)^{\frac{1}{5}}$

Simplifying the Expression

Now, we can simplify the expression using the property $(a^m)^n = a^{m \cdot n}$ :

$(13^3)^{\frac{1}{5}} = 13^{3 \cdot \frac{1}{5}}$

Evaluating the Exponent

Next, we need to evaluate the exponent $3 \cdot \frac{1}{5}$ :

$3 \cdot \frac{1}{5} = \frac{3}{5}$

Writing the Final Expression

Therefore, the final expression is:

$13^{\frac{3}{5}}$

Comparing with the Options

Now, let's compare the final expression with the given options:

A. $13^2$ B. $_{13}^{15}$ C. $13^{\frac{5}{3}}$ D. $13^{\frac{3}{5}}$

Conclusion

Based on the comparison, we can see that option D is the correct answer, as it matches the final expression we obtained.

Final Answer

The final answer is $\boxed{D}$ .

Understanding the Concept

In this problem, we applied the properties of exponents and roots to simplify the given expression. We used the property $\sqrt[n]{a} = a^{\frac{1}{n}}$ to rewrite the expression, and then applied the property $(a^m)^n = a^{m \cdot n}$ to simplify it further. This problem demonstrates the importance of understanding the properties of exponents and roots in solving mathematical expressions.

Real-World Applications

The concept of exponents and roots has numerous real-world applications, including:

Finance: Exponents and roots are used to calculate interest rates, investments, and loans.
Science: Exponents and roots are used to describe the growth and decay of populations, chemical reactions, and physical phenomena.
Engineering: Exponents and roots are used to design and optimize systems, such as electrical circuits, mechanical systems, and computer algorithms.

Conclusion

In conclusion, the problem $\sqrt[5]{13^3}$ requires the application of the properties of exponents and roots to simplify the expression. The final answer is $\boxed{D}$ , which matches the expression $13^{\frac{3}{5}}$ . This problem demonstrates the importance of understanding the properties of exponents and roots in solving mathematical expressions, and highlights the real-world applications of this concept.

Understanding Exponents and Roots

Exponents and roots are fundamental concepts in mathematics that are used to describe the power or magnitude of a number. In this article, we will answer some frequently asked questions about exponents and roots, and provide examples to illustrate the concepts.

Q: What is an exponent?

A: An exponent is a small number that is written above and to the right of a base number, indicating how many times the base number should be multiplied by itself. For example, in the expression $2^3$ , the exponent 3 indicates that the base number 2 should be multiplied by itself 3 times: $2^3 = 2 \cdot 2 \cdot 2 = 8$ .

Q: What is a root?

A: A root is the inverse operation of an exponent. It is a number that, when raised to a certain power, gives a specified value. For example, in the expression $\sqrt{16}$ , the root is 4, because $4^2 = 16$ .

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

A: To simplify an expression with a root, you can use the property $\sqrt[n]{a} = a^{\frac{1}{n}}$ . For example, to simplify the expression $\sqrt[3]{27}$ , you can rewrite it as $(27)^{\frac{1}{3}}$ , which is equal to $3$ .

Q: How do I evaluate an expression with an exponent?

A: To evaluate an expression with an exponent, you can use the property $(a^m)^n = a^{m \cdot n}$ . For example, to evaluate the expression $(2^3)^2$ , you can rewrite it as $2^{3 \cdot 2}$ , which is equal to $2^6$ .

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

A: A positive exponent indicates that the base number should be multiplied by itself a certain number of times. A negative exponent indicates that the base number should be divided by itself a certain number of times. For example, in the expression $2^{-3}$ , the negative exponent indicates that the base number 2 should be divided by itself 3 times: $2^{-3} = \frac{1}{2 \cdot 2 \cdot 2} = \frac{1}{8}$ .

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

A: To simplify an expression with a negative exponent, you can use the property $a^{-n} = \frac{1}{a^n}$ . For example, to simplify the expression $2^{-3}$ , you can rewrite it as $\frac{1}{2^3}$ , which is equal to $\frac{1}{8}$ .

Q: What is the difference between a rational exponent and an irrational exponent?

A: A rational exponent is a fraction that represents the power to which a number should be raised. An irrational exponent is a number that cannot be expressed as a fraction. For example, in the expression $2^{\frac{3}{4}}$ , the exponent is rational. In the expression $2^{\sqrt{2}}$ , the exponent is irrational.

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

A: To simplify an expression with a rational exponent, you can use the property $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ . For example, to simplify the expression $2^{\frac{3}{4}}$ , you can rewrite it as $\sqrt[4]{2^3}$ , which is equal to $\sqrt[4]{8}$ .

Q: How do I simplify an expression with an irrational exponent?

A: To simplify an expression with an irrational exponent, you cannot simplify it further using the properties of exponents and roots. For example, the expression $2^{\sqrt{2}}$ cannot be simplified further.

Conclusion

In conclusion, exponents and roots are fundamental concepts in mathematics that are used to describe the power or magnitude of a number. By understanding the properties of exponents and roots, you can simplify expressions and evaluate them correctly. Remember to use the properties of exponents and roots to simplify expressions with rational and irrational exponents.

Real-World Applications

The concept of exponents and roots has numerous real-world applications, including:

Finance: Exponents and roots are used to calculate interest rates, investments, and loans.
Science: Exponents and roots are used to describe the growth and decay of populations, chemical reactions, and physical phenomena.
Engineering: Exponents and roots are used to design and optimize systems, such as electrical circuits, mechanical systems, and computer algorithms.

Final Tips

Practice, practice, practice: The more you practice simplifying expressions with exponents and roots, the more comfortable you will become with the concepts.
Use the properties of exponents and roots: Remember to use the properties of exponents and roots to simplify expressions and evaluate them correctly.
Understand the difference between rational and irrational exponents: Remember that rational exponents can be simplified using the properties of exponents and roots, while irrational exponents cannot be simplified further.