Understanding the Properties of Radicals and Exponents

Oct 13, 2025 by miodeya.com 45 views

Introduction

In this article, we will simplify the given expression $\sqrt{20x^5y} \times \sqrt{4xy^6}$ using the properties of radicals and exponents. The expression involves the multiplication of two square roots, and we will use the rules of exponents and radicals to simplify it.

Understanding the Properties of Radicals and Exponents

Before we start simplifying the expression, let's review the properties of radicals and exponents that we will use.

The product rule for radicals states that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ .
The power rule for exponents states that $(a^m)^n = a^{mn}$ .
The rule for multiplying exponents with the same base states that $a^m \times a^n = a^{m+n}$ .

Simplifying the Expression

Now that we have reviewed the properties of radicals and exponents, let's simplify the expression $\sqrt{20x^5y} \times \sqrt{4xy^6}$ .

First, we can rewrite the expression as $\sqrt{20x^5y} \times \sqrt{4xy^6} = \sqrt{20x^5y \times 4xy^6}$ .

Next, we can simplify the expression inside the square root by multiplying the numbers and adding the exponents.

$\sqrt{20x^5y \times 4xy^6} = \sqrt{80x^6y^7}$

Simplifying the Number Inside the Square Root

Now that we have simplified the expression inside the square root, let's simplify the number inside the square root.

The number 80 can be factored as $16 \times 5$ , and the number 16 is a perfect square.

$\sqrt{80x^6y^7} = \sqrt{16 \times 5 \times x^6y^7}$

Simplifying the Expression Further

Now that we have simplified the number inside the square root, let's simplify the expression further.

$\sqrt{16 \times 5 \times x^6y^7} = \sqrt{16} \times \sqrt{5} \times \sqrt{x^6} \times \sqrt{y^7}$

Applying the Power Rule for Exponents

Now that we have simplified the expression, let's apply the power rule for exponents.

$\sqrt{16} \times \sqrt{5} \times \sqrt{x^6} \times \sqrt{y^7} = 4 \times \sqrt{5} \times x^3 \times y^3 \times \sqrt{y}$

Simplifying the Expression Further

Now that we have applied the power rule for exponents, let's simplify the expression further.

$4 \times \sqrt{5} \times x^3 \times y^3 \times \sqrt{y} = 4x^3y^3\sqrt{5y}$

Conclusion

In this article, we simplified the expression $\sqrt{20x^5y} \times \sqrt{4xy^6}$ using the properties of radicals and exponents. We reviewed the properties of radicals and exponents, simplified the expression inside the square root, simplified the number inside the square root, applied the power rule for exponents, and simplified the expression further.

The final simplified expression is $4x^3y^3\sqrt{5y}$ .

Frequently Asked Questions

What is the product rule for radicals?
- The product rule for radicals states that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ .
What is the power rule for exponents?
- The power rule for exponents states that $(a^m)^n = a^{mn}$ .
What is the rule for multiplying exponents with the same base?
- The rule for multiplying exponents with the same base states that $a^m \times a^n = a^{m+n}$ .

Final Answer

The final answer is $\boxed{4x^3y^3\sqrt{5y}}$ .

Introduction

In our previous article, we simplified the expression $\sqrt{20x^5y} \times \sqrt{4xy^6}$ using the properties of radicals and exponents. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q1: What is the product rule for radicals?

A1: The product rule for radicals states that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ .

Q2: What is the power rule for exponents?

A2: The power rule for exponents states that $(a^m)^n = a^{mn}$ .

Q3: What is the rule for multiplying exponents with the same base?

A3: The rule for multiplying exponents with the same base states that $a^m \times a^n = a^{m+n}$ .

Q4: How do I simplify the expression inside the square root?

A4: To simplify the expression inside the square root, you can multiply the numbers and add the exponents.

Q5: What is the difference between a perfect square and a non-perfect square?

A5: A perfect square is a number that can be expressed as the square of an integer, while a non-perfect square is a number that cannot be expressed as the square of an integer.

Q6: How do I simplify the number inside the square root?

A6: To simplify the number inside the square root, you can factor the number into its prime factors and look for perfect squares.

Q7: What is the final simplified expression?

A7: The final simplified expression is $4x^3y^3\sqrt{5y}$ .

Q8: Can I use the product rule for radicals to simplify the expression?

A8: Yes, you can use the product rule for radicals to simplify the expression. However, you need to make sure that the expressions inside the square roots are simplified first.

Q9: Can I use the power rule for exponents to simplify the expression?

A9: Yes, you can use the power rule for exponents to simplify the expression. However, you need to make sure that the exponents are simplified first.

Q10: What is the final answer?

A10: The final answer is $\boxed{4x^3y^3\sqrt{5y}}$ .

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $\sqrt{20x^5y} \times \sqrt{4xy^6}$ . We reviewed the product rule for radicals, the power rule for exponents, and the rule for multiplying exponents with the same base. We also provided examples and explanations to help clarify the concepts.

Frequently Asked Questions

What is the product rule for radicals?
- The product rule for radicals states that $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ .
What is the power rule for exponents?
- The power rule for exponents states that $(a^m)^n = a^{mn}$ .
What is the rule for multiplying exponents with the same base?
- The rule for multiplying exponents with the same base states that $a^m \times a^n = a^{m+n}$ .

Final Answer

The final answer is $\boxed{4x^3y^3\sqrt{5y}}$ .