Simplify The Expression:$\sqrt{2}(4 \sqrt{10} - 6 \sqrt{6}$\]

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. When dealing with expressions involving square roots, it's essential to apply the correct techniques to simplify them. In this article, we will focus on simplifying the given expression: $\sqrt{2}(4 \sqrt{10} - 6 \sqrt{6})$. We will use various mathematical techniques, including the distributive property and the properties of square roots, to simplify the expression.

Understanding the Expression

The given expression is $\sqrt{2}(4 \sqrt{10} - 6 \sqrt{6})$. This expression involves the product of a square root and a binomial. To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We will use this property to expand the expression and simplify it further.

Applying the Distributive Property

To simplify the expression, we will apply the distributive property by multiplying the square root with each term inside the parentheses. This will give us:

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

Simplifying the Terms

Now that we have expanded the expression, we can simplify each term separately. We will start by simplifying the first term, $\sqrt{2} \cdot 4 \sqrt{10}$. To simplify this term, we can use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. Applying this property, we get:

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

Further Simplification

We can further simplify the term $4 \sqrt{20}$ by expressing $20$ as a product of its prime factors. This will give us:

$4 \sqrt{20} = 4 \sqrt{2^2 \cdot 5} = 4 \cdot 2 \sqrt{5} = 8 \sqrt{5}$

Simplifying the Second Term

Now that we have simplified the first term, we can simplify the second term, $\sqrt{2} \cdot 6 \sqrt{6}$. To simplify this term, we can use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. Applying this property, we get:

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

Further Simplification

We can further simplify the term $6 \sqrt{12}$ by expressing $12$ as a product of its prime factors. This will give us:

$6 \sqrt{12} = 6 \sqrt{2^2 \cdot 3} = 6 \cdot 2 \sqrt{3} = 12 \sqrt{3}$

Combining the Terms

Now that we have simplified both terms, we can combine them to get the final simplified expression. We will subtract the second term from the first term:

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

Conclusion

In this article, we simplified the given expression $\sqrt{2}(4 \sqrt{10} - 6 \sqrt{6})$ using the distributive property and the properties of square roots. We expanded the expression, simplified each term separately, and combined the terms to get the final simplified expression. The simplified expression is $8 \sqrt{5} - 12 \sqrt{3}$.

Final Answer

The final answer is $\boxed{8 \sqrt{5} - 12 \sqrt{3}}$.

Related Topics

• Simplifying expressions involving square roots
• Distributive property
• Properties of square roots
• Simplifying binomials

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for general information purposes only and are not directly related to the specific problem discussed in this article.

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Introduction

In our previous article, we simplified the expression $\sqrt{2}(4 \sqrt{10} - 6 \sqrt{6})$ using the distributive property and the properties of square roots. In this article, we will answer some frequently asked questions related to the simplification of this expression.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical technique that allows us to expand an expression by multiplying each term inside the parentheses with the term outside the parentheses. In the case of the expression $\sqrt{2}(4 \sqrt{10} - 6 \sqrt{6})$, we used the distributive property to expand the expression as $\sqrt{2} \cdot 4 \sqrt{10} - \sqrt{2} \cdot 6 \sqrt{6}$.

Q: How do we simplify the terms inside the parentheses?

A: To simplify the terms inside the parentheses, we can use the properties of square roots. For example, to simplify the term $\sqrt{2} \cdot 4 \sqrt{10}$, we can use the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ to get $4 \sqrt{2 \cdot 10} = 4 \sqrt{20}$.

Q: How do we simplify the expression $4 \sqrt{20}$?

A: To simplify the expression $4 \sqrt{20}$, we can express $20$ as a product of its prime factors, which is $2^2 \cdot 5$. Then, we can use the property $\sqrt{a^2} = a$ to simplify the expression as $4 \cdot 2 \sqrt{5} = 8 \sqrt{5}$.

Q: How do we simplify the expression $6 \sqrt{12}$?

A: To simplify the expression $6 \sqrt{12}$, we can express $12$ as a product of its prime factors, which is $2^2 \cdot 3$. Then, we can use the property $\sqrt{a^2} = a$ to simplify the expression as $6 \cdot 2 \sqrt{3} = 12 \sqrt{3}$.

Q: What is the final simplified expression?

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

Q: How do we combine the terms?

A: To combine the terms, we can simply subtract the second term from the first term: $8 \sqrt{5} - 12 \sqrt{3}$.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $\sqrt{2}(4 \sqrt{10} - 6 \sqrt{6})$. We used the distributive property and the properties of square roots to simplify the expression and provided step-by-step solutions to each question.

Final Answer

The final answer is $\boxed{8 \sqrt{5} - 12 \sqrt{3}}$.

Related Topics

• Simplifying expressions involving square roots
• Distributive property
• Properties of square roots
• Simplifying binomials

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Note: The references provided are for general information purposes only and are not directly related to the specific problem discussed in this article.