Understanding the Properties of Radicals

Oct 13, 2025 by windailys.com 59 views

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Introduction

In this article, we will simplify the given expression $\sqrt{2}(4 \sqrt{10} - 6 \sqrt{6})$ . This involves using the properties of radicals and simplifying the expression to its simplest form. We will also compare the simplified expression with the given options to determine the correct answer.

Understanding the Properties of Radicals

Before we begin simplifying the expression, it's essential to understand the properties of radicals. The properties of radicals include:

Product Property: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$
Quotient Property: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
Power Property: $(\sqrt{a})^n = a^{\frac{n}{2}}$

Simplifying the Expression

To simplify the expression $\sqrt{2}(4 \sqrt{10} - 6 \sqrt{6})$ , we can use the distributive property to multiply the square root of 2 with each term inside the parentheses.

$\sqrt{2}(4 \sqrt{10} - 6 \sqrt{6}) = 4\sqrt{2} \cdot \sqrt{10} - 6\sqrt{2} \cdot \sqrt{6}$

Using the product property of radicals, we can simplify each term:

$4\sqrt{2} \cdot \sqrt{10} = 4\sqrt{2 \cdot 10} = 4\sqrt{20}$

$6\sqrt{2} \cdot \sqrt{6} = 6\sqrt{2 \cdot 6} = 6\sqrt{12}$

Combining Like Terms

Now that we have simplified each term, we can combine like terms to simplify the expression further.

$4\sqrt{20} - 6\sqrt{12}$

We can simplify the square roots of 20 and 12 by factoring out perfect squares.

$\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$

$\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$

Substituting these values back into the expression, we get:

$4(2\sqrt{5}) - 6(2\sqrt{3})$

Using the distributive property, we can simplify each term:

$8\sqrt{5} - 12\sqrt{3}$

Comparing with the Given Options

Now that we have simplified the expression, we can compare it with the given options to determine the correct answer.

Option A: $4 \sqrt{20} - 6 \sqrt{12}$

Option B: $4 \sqrt{12} - 6 \sqrt{8}$

Option C: $8 \sqrt{5} - 12 \sqrt{3}$

Option D: $8 \sqrt{10} - 6 \sqrt{8}$

Comparing the simplified expression with each option, we can see that option C matches the simplified expression.

Conclusion

In this article, we simplified the given expression $\sqrt{2}(4 \sqrt{10} - 6 \sqrt{6})$ using the properties of radicals and the distributive property. We also compared the simplified expression with the given options to determine the correct answer. The correct answer is option C: $8 \sqrt{5} - 12 \sqrt{3}$ .

Frequently Asked Questions

Q: What is the property of radicals used to simplify the expression?

A: The product property of radicals is used to simplify the expression.

Q: How do you simplify the square root of 20?

A: The square root of 20 can be simplified by factoring out a perfect square: $\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$ .

Q: How do you simplify the square root of 12?

A: The square root of 12 can be simplified by factoring out a perfect square: $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$ .

Q: What is the correct answer?

A: The correct answer is option C: $8 \sqrt{5} - 12 \sqrt{3}$ .

References

Q&A: Simplifying Radical Expressions

Q: What is the property of radicals used to simplify the expression?

A: The product property of radicals is used to simplify the expression. This property states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ .

Q: How do you simplify the square root of 20?

A: The square root of 20 can be simplified by factoring out a perfect square: $\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$ .

Q: How do you simplify the square root of 12?

A: The square root of 12 can be simplified by factoring out a perfect square: $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$ .

Q: What is the correct answer?

A: The correct answer is option C: $8 \sqrt{5} - 12 \sqrt{3}$ .

Q: Can you explain the distributive property?

A: The distributive property is a mathematical property that allows us to distribute a single value to multiple values. In the context of simplifying radical expressions, the distributive property is used to multiply a single value to multiple values inside the parentheses.

Q: How do you use the distributive property to simplify the expression?

A: To simplify the expression using the distributive property, we multiply the value outside the parentheses to each value inside the parentheses. In this case, we multiply $\sqrt{2}$ to each value inside the parentheses: $4\sqrt{2} \cdot \sqrt{10} - 6\sqrt{2} \cdot \sqrt{6}$ .

Q: Can you explain the concept of combining like terms?

A: Combining like terms is a mathematical concept that involves combining two or more terms that have the same variable and coefficient. In the context of simplifying radical expressions, combining like terms involves combining two or more terms that have the same radical expression.

Q: How do you combine like terms in the expression?

A: To combine like terms in the expression, we simplify each term by factoring out perfect squares and then combine the like terms. In this case, we simplify each term by factoring out perfect squares: $4\sqrt{20} - 6\sqrt{12}$ , and then combine the like terms: $8\sqrt{5} - 12\sqrt{3}$ .

Q: What is the final simplified expression?

A: The final simplified expression is $8\sqrt{5} - 12\sqrt{3}$ .

Common Mistakes to Avoid

Q: What is a common mistake to avoid when simplifying radical expressions?

A: A common mistake to avoid when simplifying radical expressions is to forget to factor out perfect squares.

Q: How do you avoid this mistake?

A: To avoid this mistake, make sure to factor out perfect squares from each term in the expression.

Q: What is another common mistake to avoid?

A: Another common mistake to avoid is to forget to combine like terms.

Q: How do you avoid this mistake?

A: To avoid this mistake, make sure to combine like terms after simplifying each term.

Tips and Tricks

Q: What is a tip for simplifying radical expressions?

A: A tip for simplifying radical expressions is to use the product property of radicals to simplify each term.

Q: How do you use the product property of radicals?

A: To use the product property of radicals, multiply the value outside the parentheses to each value inside the parentheses.

Q: What is another tip for simplifying radical expressions?

A: Another tip for simplifying radical expressions is to factor out perfect squares from each term.

Q: How do you factor out perfect squares?

A: To factor out perfect squares, look for perfect squares inside the radical expression and factor them out.

Conclusion

Frequently Asked Questions

Q: What is the property of radicals used to simplify the expression?

A: The product property of radicals is used to simplify the expression.

Q: How do you simplify the square root of 20?

A: The square root of 20 can be simplified by factoring out a perfect square: $\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$ .

Q: How do you simplify the square root of 12?

A: The square root of 12 can be simplified by factoring out a perfect square: $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$ .

Q: What is the correct answer?

A: The correct answer is option C: $8 \sqrt{5} - 12 \sqrt{3}$ .