Find The Sum:${ 4 \sqrt{3} + 11 \sqrt{12} }$A. { 15 \sqrt{15} $}$B. { 15 \sqrt{3} $}$C. { 26 \sqrt{3} $}$D. { 48 \sqrt{3} $}$

Mar 11, 2025 by ADMIN 127 views

**Simplifying Radical Expressions: A Step-by-Step Guide**

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Understanding the Problem

When dealing with radical expressions, it's essential to simplify them to make calculations easier. In this article, we'll focus on simplifying the given expression: $4 \sqrt{3} + 11 \sqrt{12}$ . Our goal is to find the sum of these two radical expressions.

Breaking Down the Expression

To simplify the expression, we need to break it down into its components. The first term is $4 \sqrt{3}$ , and the second term is $11 \sqrt{12}$ . We can start by simplifying the second term, which involves finding the prime factorization of $12$ .

Prime Factorization of 12

The prime factorization of $12$ is $2^2 \cdot 3$ . This means that $\sqrt{12}$ can be rewritten as $\sqrt{2^2 \cdot 3}$ .

Simplifying the Second Term

Using the property of radicals that allows us to split the square root of a product into the product of square roots, we can rewrite $\sqrt{12}$ as $\sqrt{2^2} \cdot \sqrt{3}$ . This simplifies to $2 \sqrt{3}$ .

Rewriting the Expression

Now that we've simplified the second term, we can rewrite the original expression as $4 \sqrt{3} + 11 (2 \sqrt{3})$ .

Combining Like Terms

We can combine the like terms in the expression by multiplying $11$ by $2 \sqrt{3}$ , which gives us $22 \sqrt{3}$ . The expression now becomes $4 \sqrt{3} + 22 \sqrt{3}$ .

Final Simplification

The final step is to combine the like terms by adding $4 \sqrt{3}$ and $22 \sqrt{3}$ . This gives us a simplified expression of $26 \sqrt{3}$ .

Conclusion

In this article, we simplified the given radical expression $4 \sqrt{3} + 11 \sqrt{12}$ by breaking it down into its components, simplifying the second term, and combining like terms. The final simplified expression is $26 \sqrt{3}$ .

Answer

The correct answer is:

C. $26 \sqrt{3}$

Why This Matters

Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. By simplifying expressions, we can make calculations easier and more efficient. This skill is also crucial in real-world applications, such as engineering and physics, where radical expressions are often used to describe complex phenomena.

Real-World Applications

Radical expressions are used in various real-world applications, including:

Engineering: Radical expressions are used to describe the dimensions of complex shapes, such as bridges and buildings.
Physics: Radical expressions are used to describe the motion of objects, such as the trajectory of a projectile.
Computer Science: Radical expressions are used in algorithms and data structures to optimize performance and efficiency.

Tips and Tricks

Here are some tips and tricks for simplifying radical expressions:

Use the property of radicals: The property of radicals allows us to split the square root of a product into the product of square roots.
Simplify the radicand: Simplify the radicand by finding the prime factorization of the number inside the square root.
Combine like terms: Combine like terms by adding or subtracting the coefficients of the radical expressions.

Common Mistakes

Here are some common mistakes to avoid when simplifying radical expressions:

Not simplifying the radicand: Failing to simplify the radicand can lead to incorrect results.
Not combining like terms: Failing to combine like terms can lead to incorrect results.
Not using the property of radicals: Failing to use the property of radicals can lead to incorrect results.

Conclusion

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Frequently Asked Questions

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or other root. It is a way of representing a number that is the result of a root operation.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow these steps:

Simplify the radicand: Simplify the radicand by finding the prime factorization of the number inside the square root.
Use the property of radicals: Use the property of radicals to split the square root of a product into the product of square roots.
Combine like terms: Combine like terms by adding or subtracting the coefficients of the radical expressions.

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

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., it can be written in the form a/b, where a and b are integers and b is not equal to zero. An irrational number is a number that cannot be expressed as the ratio of two integers.

Q: How do I simplify a square root of a product?

A: To simplify a square root of a product, you need to use the property of radicals, which states that the square root of a product is equal to the product of the square roots. For example, √(ab) = √a * √b.

Q: What is the difference between a perfect square and a perfect cube?

A: A perfect square is a number that can be expressed as the square of an integer, i.e., it can be written in the form n^2, where n is an integer. A perfect cube is a number that can be expressed as the cube of an integer, i.e., it can be written in the form n^3, where n is an integer.

Q: How do I simplify a radical expression with a coefficient?

A: To simplify a radical expression with a coefficient, you need to follow these steps:

Simplify the radicand: Simplify the radicand by finding the prime factorization of the number inside the square root.
Use the property of radicals: Use the property of radicals to split the square root of a product into the product of square roots.
Combine like terms: Combine like terms by adding or subtracting the coefficients of the radical expressions.

Q: What is the difference between a radical expression and an algebraic expression?

A: A radical expression is an expression that contains a square root or other root. An algebraic expression is a general term that refers to any expression that involves variables and constants, including radical expressions.

Q: How do I simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, you need to follow these steps:

Simplify the radicand: Simplify the radicand by finding the prime factorization of the number inside the square root.
Use the property of radicals: Use the property of radicals to split the square root of a product into the product of square roots.
Combine like terms: Combine like terms by adding or subtracting the coefficients of the radical expressions.

Q: What is the difference between a radical expression and a polynomial expression?

A: A radical expression is an expression that contains a square root or other root. A polynomial expression is a general term that refers to any expression that involves variables and constants, including radical expressions, but with the restriction that the variables are raised to non-negative integer powers.

Q: How do I simplify a radical expression with a polynomial?

A: To simplify a radical expression with a polynomial, you need to follow these steps:

Simplify the radicand: Simplify the radicand by finding the prime factorization of the number inside the square root.
Use the property of radicals: Use the property of radicals to split the square root of a product into the product of square roots.
Combine like terms: Combine like terms by adding or subtracting the coefficients of the radical expressions.

Conclusion

Simplifying radical expressions is an essential skill in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, you can simplify radical expressions with confidence. Remember to simplify the radicand, use the property of radicals, and combine like terms to get the final result.