Applying the Properties of Exponents and Roots

Oct 13, 2025 by magicalceylontours.com 20 views

Understanding the Problem

The given expression involves the multiplication of two cube roots, each raised to a different power of 10. To simplify this expression, we need to apply the properties of exponents and roots. The expression $\sqrt[3]{10^2} \cdot \sqrt[3]{10^4}$ can be rewritten using the properties of exponents and roots, and then simplified to a single expression.

Applying the Properties of Exponents and Roots

The expression $\sqrt[3]{10^2} \cdot \sqrt[3]{10^4}$ can be rewritten as $\sqrt[3]{10^2 \cdot 10^4}$ . This is because the cube root of a product is equal to the product of the cube roots. Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$ , we can simplify the expression inside the cube root.

Simplifying the Expression Inside the Cube Root

Using the property of exponents that states $a^m \cdot a^n = a^{m+n}$ , we can simplify the expression inside the cube root as follows:

10^2 \cdot 10^4 = 10^{2+4} = 10^6

Therefore, the expression $\sqrt[3]{10^2} \cdot \sqrt[3]{10^4}$ can be rewritten as $\sqrt[3]{10^6}$ .

Simplifying the Cube Root

To simplify the cube root, we can use the property of exponents that states $\sqrt[n]{a^n} = a$ . In this case, we have $\sqrt[3]{10^6}$ , which can be rewritten as $10^{\frac{6}{3}}$ .

Simplifying the Fractional Exponent

Using the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ , we can simplify the fractional exponent as follows:

10^{\frac{6}{3}} = \sqrt[3]{10^6} = 10^2

Therefore, the expression $\sqrt[3]{10^2} \cdot \sqrt[3]{10^4}$ is equivalent to $10^2$ .

Conclusion

In conclusion, the expression $\sqrt[3]{10^2} \cdot \sqrt[3]{10^4}$ can be simplified using the properties of exponents and roots. By applying the properties of exponents and roots, we can rewrite the expression as $\sqrt[3]{10^6}$ , and then simplify the cube root to $10^2$ . Therefore, the correct answer is D. $10^2$ .

Frequently Asked Questions

Q: What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$ ? A: This property states that when we multiply two numbers with the same base, we can add their exponents.
Q: What is the property of exponents that states $\sqrt[n]{a^n} = a$ ? A: This property states that when we take the nth root of a number raised to the power of n, we get the original number.
Q: What is the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ ? A: This property states that when we raise a number to a fractional exponent, we can rewrite it as the nth root of the number raised to the power of m.

Step-by-Step Solution

Rewrite the expression $\sqrt[3]{10^2} \cdot \sqrt[3]{10^4}$ as $\sqrt[3]{10^2 \cdot 10^4}$ .
Simplify the expression inside the cube root using the property of exponents that states $a^m \cdot a^n = a^{m+n}$ .
Rewrite the expression inside the cube root as $10^6$ .
Simplify the cube root using the property of exponents that states $\sqrt[n]{a^n} = a$ .
Rewrite the cube root as $10^{\frac{6}{3}}$ .
Simplify the fractional exponent using the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ .
Rewrite the fractional exponent as $10^2$ .

Final Answer

The final answer is D. $10^2$ .

Understanding Exponents and Roots

Exponents and roots are fundamental concepts in mathematics that help us simplify complex expressions. In the previous article, we explored the expression $\sqrt[3]{10^2} \cdot \sqrt[3]{10^4}$ and simplified it to $10^2$ . In this article, we will answer some frequently asked questions about exponents and roots.

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

A: An exponent is a number that is raised to a power, while a root is the inverse operation of an exponent. For example, $10^2$ is an exponent, while $\sqrt{10^2}$ is a root.

Q: What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$ ?

A: This property states that when we multiply two numbers with the same base, we can add their exponents. For example, $10^2 \cdot 10^3 = 10^{2+3} = 10^5$ .

Q: What is the property of exponents that states $\sqrt[n]{a^n} = a$ ?

A: This property states that when we take the nth root of a number raised to the power of n, we get the original number. For example, $\sqrt[3]{10^3} = 10$ .

Q: What is the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ ?

A: This property states that when we raise a number to a fractional exponent, we can rewrite it as the nth root of the number raised to the power of m. For example, $10^{\frac{2}{3}} = \sqrt[3]{10^2}$ .

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

A: To simplify an expression with multiple exponents, you can use the properties of exponents to combine the exponents. For example, $10^2 \cdot 10^3 = 10^{2+3} = 10^5$ .

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

A: To simplify an expression with multiple roots, you can use the properties of roots to combine the roots. For example, $\sqrt{10^2} \cdot \sqrt{10^3} = \sqrt{10^2 \cdot 10^3} = \sqrt{10^5}$ .

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

A: A rational exponent is an exponent that can be expressed as a fraction, while an irrational exponent is an exponent that cannot be expressed as a fraction. For example, $10^{\frac{1}{2}}$ is a rational exponent, while $10^{\sqrt{2}}$ is an irrational exponent.

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 of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ . For example, $10^{\frac{1}{2}} = \sqrt{10}$ .

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. For example, $10^{\sqrt{2}}$ cannot be simplified further.

Conclusion

In conclusion, exponents and roots are fundamental concepts in mathematics that help us simplify complex expressions. By understanding the properties of exponents and roots, we can simplify expressions with multiple exponents and roots. We hope this Q&A article has helped you understand exponents and roots better.

Frequently Asked Questions

Q: What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$ ? A: This property states that when we multiply two numbers with the same base, we can add their exponents.
Q: What is the property of exponents that states $\sqrt[n]{a^n} = a$ ? A: This property states that when we take the nth root of a number raised to the power of n, we get the original number.
Q: What is the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ ? A: This property states that when we raise a number to a fractional exponent, we can rewrite it as the nth root of the number raised to the power of m.

Step-by-Step Solution

Understand the properties of exponents and roots.
Simplify expressions with multiple exponents using the properties of exponents.
Simplify expressions with multiple roots using the properties of roots.
Understand the difference between rational and irrational exponents.
Simplify expressions with rational exponents using the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$ .
Understand that expressions with irrational exponents cannot be simplified further.

Final Answer

The final answer is that exponents and roots are fundamental concepts in mathematics that help us simplify complex expressions. By understanding the properties of exponents and roots, we can simplify expressions with multiple exponents and roots.