Which Choice Is Equivalent To The Expression Below? 6 + 18 + 3 3 − 3 2 \sqrt{6}+\sqrt{18}+3 \sqrt{3}-3 \sqrt{2} 6 ​ + 18 ​ + 3 3 ​ − 3 2 ​ A. 3 3 + 6 3 \sqrt{3}+\sqrt{6} 3 3 ​ + 6 ​ B. 5 3 5 \sqrt{3} 5 3 ​ C. 2 3 − 24 2 \sqrt{3}-\sqrt{24} 2 3 ​ − 24 ​ D. 5 3 − 6 5 \sqrt{3}-6 5 3 ​ − 6

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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 explore the process of simplifying radical expressions, with a focus on the given expression: 6+18+33−32\sqrt{6}+\sqrt{18}+3 \sqrt{3}-3 \sqrt{2}. We will examine each option and determine which one is equivalent to the given expression.

Understanding Radical Expressions

Before we dive into the simplification process, let's take a moment to understand what radical expressions are. A radical expression is any expression that contains a square root or other root. 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 the Given Expression

Now that we have a basic understanding of radical expressions, let's simplify the given expression: 6+18+33−32\sqrt{6}+\sqrt{18}+3 \sqrt{3}-3 \sqrt{2}. To simplify this expression, we need to combine like terms and simplify each radical.

Step 1: Simplify the Radicals

The first step in simplifying the given expression is to simplify each radical. We can start by simplifying 18\sqrt{18}. We can rewrite 18\sqrt{18} as 9×2\sqrt{9 \times 2}, which simplifies to 323\sqrt{2}.

$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

Now that we have simplified 18\sqrt{18}, we can rewrite the given expression as: 6+32+33−32\sqrt{6}+3\sqrt{2}+3 \sqrt{3}-3 \sqrt{2}.

Step 2: Combine Like Terms

The next step in simplifying the given expression is to combine like terms. We can start by combining the like terms 323\sqrt{2} and −32-3 \sqrt{2}, which cancel each other out.

$3\sqrt{2} - 3\sqrt{2} = 0$

Now that we have combined the like terms, we are left with: 6+33\sqrt{6}+3 \sqrt{3}.

Step 3: Simplify the Remaining Radicals

The final step in simplifying the given expression is to simplify the remaining radicals. We can start by simplifying 6\sqrt{6}. We can rewrite 6\sqrt{6} as 2×3\sqrt{2 \times 3}, which simplifies to 23\sqrt{2}\sqrt{3}.

$\sqrt{6} = \sqrt{2 \times 3} = \sqrt{2}\sqrt{3}$

Now that we have simplified 6\sqrt{6}, we can rewrite the given expression as: 23+33\sqrt{2}\sqrt{3}+3 \sqrt{3}.

Step 4: Combine Like Terms Again

The final step in simplifying the given expression is to combine like terms again. We can start by combining the like terms 23\sqrt{2}\sqrt{3} and 333 \sqrt{3}, which can be rewritten as 33+233\sqrt{3}+\sqrt{2}\sqrt{3}.

$3\sqrt{3}+\sqrt{2}\sqrt{3} = 3\sqrt{3}+\sqrt{6}$

Conclusion

In conclusion, the given expression 6+18+33−32\sqrt{6}+\sqrt{18}+3 \sqrt{3}-3 \sqrt{2} simplifies to 33+63\sqrt{3}+\sqrt{6}. This is the equivalent expression to the given expression.

Which Choice is Equivalent to the Expression?

Now that we have simplified the given expression, we can examine each option and determine which one is equivalent to the expression.

  • Option A: 33+63 \sqrt{3}+\sqrt{6}

    This option is equivalent to the given expression, as we simplified the expression to 33+63\sqrt{3}+\sqrt{6}.

  • Option B: 535 \sqrt{3}

    This option is not equivalent to the given expression, as we simplified the expression to 33+63\sqrt{3}+\sqrt{6}, not 535 \sqrt{3}.

  • Option C: 23−242 \sqrt{3}-\sqrt{24}

    This option is not equivalent to the given expression, as we simplified the expression to 33+63\sqrt{3}+\sqrt{6}, not 23−242 \sqrt{3}-\sqrt{24}.

  • Option D: 53−65 \sqrt{3}-6

    This option is not equivalent to the given expression, as we simplified the expression to 33+63\sqrt{3}+\sqrt{6}, not 53−65 \sqrt{3}-6.

Final Answer

Q: What is a radical expression?

A: A radical expression is any expression that contains a square root or other root. The square root of a number is a value that, when multiplied by itself, gives the original number.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to combine like terms and simplify each radical. You can start by simplifying each radical, then combine like terms, and finally simplify the remaining radicals.

Q: What is the difference between a like term and a unlike term?

A: A like term is a term that has the same variable and exponent. For example, 222\sqrt{2} and 323\sqrt{2} are like terms because they both have the variable 2\sqrt{2}. A unlike term is a term that has a different variable or exponent. For example, 222\sqrt{2} and 333\sqrt{3} are unlike terms because they have different variables.

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

A: To simplify a radical expression with multiple terms, you need to combine like terms and simplify each radical. You can start by simplifying each radical, then combine like terms, and finally simplify the remaining radicals.

Q: What is the order of operations for simplifying radical expressions?

A: The order of operations for simplifying radical expressions is:

  1. Simplify each radical
  2. Combine like terms
  3. Simplify the remaining radicals

Q: Can I simplify a radical expression with a negative number?

A: Yes, you can simplify a radical expression with a negative number. To simplify a radical expression with a negative number, you need to follow the same steps as simplifying a radical expression with a positive number.

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

A: To simplify a radical expression with a variable, you need to follow the same steps as simplifying a radical expression with a number. You can start by simplifying each radical, then combine like terms, and finally simplify the remaining radicals.

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

A: A rational expression is an expression that contains a fraction, while a radical expression is an expression that contains a square root or other root. For example, 12\frac{1}{2} is a rational expression, while 2\sqrt{2} is a radical expression.

Q: Can I simplify a radical expression with a rational expression?

A: Yes, you can simplify a radical expression with a rational expression. To simplify a radical expression with a rational expression, you need to follow the same steps as simplifying a radical expression with a number.

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

A: To simplify a radical expression with multiple variables, you need to follow the same steps as simplifying a radical expression with a single variable. You can start by simplifying each radical, then combine like terms, and finally simplify the remaining radicals.

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

A: A radical expression is an expression that contains a square root or other root, while an exponential expression is an expression that contains a power or exponent. For example, 2\sqrt{2} is a radical expression, while 232^3 is an exponential expression.

Q: Can I simplify a radical expression with an exponential expression?

A: Yes, you can simplify a radical expression with an exponential expression. To simplify a radical expression with an exponential expression, you need to follow the same steps as simplifying a radical expression with a number.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master in mathematics. By following the steps outlined in this article, you can simplify radical expressions with ease. Remember to combine like terms, simplify each radical, and finally simplify the remaining radicals. With practice and patience, you will become a pro at simplifying radical expressions in no time!