Simplify: ${ -\sqrt{48} - \sqrt{36} - \sqrt{16} }$A. { -4 \sqrt{3} + 10$}$B. ${ 10 - 4 \sqrt{3}\$} C. ${ 4 \sqrt{3} - 10\$} D. { -10 - 4 \sqrt{3}$}$

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will delve into the world of radical expressions and provide a step-by-step guide on how to simplify them. We will focus on the given expression: −48−36−16-\sqrt{48} - \sqrt{36} - \sqrt{16} and explore the different options provided.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or a higher root of a number. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Simplifying Radical Expressions

To simplify a radical expression, we need to find the largest perfect square that divides the number inside the square root. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as 4 multiplied by 4.

Step 1: Factorize the Numbers Inside the Square Roots

The first step in simplifying the given expression is to factorize the numbers inside the square roots.

  • 48\sqrt{48} can be factorized as 16×3\sqrt{16 \times 3}.
  • 36\sqrt{36} can be factorized as 36\sqrt{36}, which is a perfect square.
  • 16\sqrt{16} can be factorized as 16\sqrt{16}, which is a perfect square.

Step 2: Simplify the Square Roots

Now that we have factorized the numbers inside the square roots, we can simplify them.

  • 16×3\sqrt{16 \times 3} can be simplified as 16×3\sqrt{16} \times \sqrt{3}, which equals 434\sqrt{3}.
  • 36\sqrt{36} can be simplified as 66.
  • 16\sqrt{16} can be simplified as 44.

Step 3: Combine the Simplified Expressions

Now that we have simplified the individual square roots, we can combine them to get the final expression.

−48−36−16-\sqrt{48} - \sqrt{36} - \sqrt{16} can be simplified as −43−6−4-4\sqrt{3} - 6 - 4.

Evaluating the Options

Now that we have simplified the given expression, we can evaluate the options provided.

  • Option A: −43+10-4\sqrt{3} + 10
  • Option B: 10−4310 - 4\sqrt{3}
  • Option C: 43−104\sqrt{3} - 10
  • Option D: −10−43-10 - 4\sqrt{3}

Conclusion

In conclusion, simplifying radical expressions requires a step-by-step approach. We need to factorize the numbers inside the square roots, simplify the square roots, and combine the simplified expressions. By following these steps, we can simplify even the most complex radical expressions.

Final Answer

The final answer is: B. 10−4310 - 4 \sqrt{3}

Frequently Asked Questions

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to find the largest perfect square that divides the number inside the square root and then simplify the square root.

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the product of an integer with itself.

Q: How do I factorize a number inside a square root?

A: To factorize a number inside a square root, you need to find the largest perfect square that divides the number and then simplify the square root.

References

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In our previous article, we provided a step-by-step guide on how to simplify radical expressions. In this article, we will answer some of the most frequently asked questions about simplifying radical expressions.

Q&A

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to find the largest perfect square that divides the number inside the square root and then simplify the square root.

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the product of an integer with itself.

Q: How do I factorize a number inside a square root?

A: To factorize a number inside a square root, you need to find the largest perfect square that divides the number and then simplify the square root.

Q: Can I simplify a radical expression with a negative number inside the square root?

A: Yes, you can simplify a radical expression with a negative number inside the square root. However, you need to remember that the square root of a negative number is an imaginary number.

Q: How do I simplify a radical expression with a variable inside the square root?

A: To simplify a radical expression with a variable inside the square root, you need to find the largest perfect square that divides the variable and then simplify the square root.

Q: Can I simplify a radical expression with a fraction inside the square root?

A: Yes, you can simplify a radical expression with a fraction inside the square root. However, you need to remember that the square root of a fraction is the square root of the numerator divided by the square root of the denominator.

Q: How do I simplify a radical expression with multiple square roots?

A: To simplify a radical expression with multiple square roots, you need to simplify each square root individually and then combine the simplified expressions.

Q: Can I simplify a radical expression with a radical inside another radical?

A: Yes, you can simplify a radical expression with a radical inside another radical. However, you need to remember that the order of operations matters, and you need to simplify the inner radical first.

Tips and Tricks

Tip 1: Always look for perfect squares

When simplifying a radical expression, always look for perfect squares that divide the number inside the square root. This will help you simplify the expression quickly and efficiently.

Tip 2: Use the order of operations

When simplifying a radical expression, remember to follow the order of operations (PEMDAS):

  1. Parentheses
  2. Exponents
  3. Multiplication and Division
  4. Addition and Subtraction

Tip 3: Simplify the inner radical first

When simplifying a radical expression with multiple square roots, simplify the inner radical first. This will help you simplify the expression quickly and efficiently.

Conclusion

Simplifying radical expressions requires a step-by-step approach and a good understanding of the concepts involved. By following the tips and tricks provided in this article, you can simplify even the most complex radical expressions.

Final Answer

The final answer is: There is no final answer, as this is a Q&A article.

Frequently Asked Questions

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to find the largest perfect square that divides the number inside the square root and then simplify the square root.

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the product of an integer with itself.

Q: How do I factorize a number inside a square root?

A: To factorize a number inside a square root, you need to find the largest perfect square that divides the number and then simplify the square root.

References