Simplify The Expression:$\[ -2 \sqrt{18} - 4 \sqrt{8} + 4 \sqrt{27} + 4 \sqrt{128} \\]

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Introduction

Radical expressions can be complex and intimidating, but with the right techniques, they can be simplified to reveal their underlying structure. In this article, we will explore the process of simplifying radical expressions, focusing on the given expression: βˆ’218βˆ’48+427+4128-2 \sqrt{18} - 4 \sqrt{8} + 4 \sqrt{27} + 4 \sqrt{128}. We will break down the expression into manageable parts, simplify each radical, and then combine the results to obtain the final simplified expression.

Understanding Radical Expressions

Before we dive into simplifying the given expression, let's take a moment to understand what radical expressions are and how they work. A radical expression is a mathematical expression that contains a square root or other root of a number. The radical symbol, \sqrt{}, is used to indicate that the expression inside the symbol is to be taken as a root. For example, 16\sqrt{16} is read as "the square root of 16".

Radical expressions can be simplified by factoring out perfect squares from under the radical sign. 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 4Γ—44 \times 4. When we factor out a perfect square from under the radical sign, we can simplify the expression.

Simplifying the Given Expression

Now that we have a basic understanding of radical expressions, let's apply this knowledge to simplify the given expression: βˆ’218βˆ’48+427+4128-2 \sqrt{18} - 4 \sqrt{8} + 4 \sqrt{27} + 4 \sqrt{128}.

Step 1: Simplify 18\sqrt{18}

The first step in simplifying the given expression is to simplify 18\sqrt{18}. We can do this by factoring out a perfect square from under the radical sign. We know that 18=9Γ—218 = 9 \times 2, and since 9 is a perfect square, we can rewrite 18\sqrt{18} as 9Γ—2\sqrt{9 \times 2}.

18=9Γ—2=9Γ—2=32\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3 \sqrt{2}

Step 2: Simplify 8\sqrt{8}

Next, we simplify 8\sqrt{8}. We can do this by factoring out a perfect square from under the radical sign. We know that 8=4Γ—28 = 4 \times 2, and since 4 is a perfect square, we can rewrite 8\sqrt{8} as 4Γ—2\sqrt{4 \times 2}.

8=4Γ—2=4Γ—2=22\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2 \sqrt{2}

Step 3: Simplify 27\sqrt{27}

Now, we simplify 27\sqrt{27}. We can do this by factoring out a perfect square from under the radical sign. We know that 27=9Γ—327 = 9 \times 3, and since 9 is a perfect square, we can rewrite 27\sqrt{27} as 9Γ—3\sqrt{9 \times 3}.

27=9Γ—3=9Γ—3=33\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3 \sqrt{3}

Step 4: Simplify 128\sqrt{128}

Finally, we simplify 128\sqrt{128}. We can do this by factoring out a perfect square from under the radical sign. We know that 128=64Γ—2128 = 64 \times 2, and since 64 is a perfect square, we can rewrite 128\sqrt{128} as 64Γ—2\sqrt{64 \times 2}.

128=64Γ—2=64Γ—2=82\sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8 \sqrt{2}

Combining the Simplified Radicals

Now that we have simplified each radical, we can combine the results to obtain the final simplified expression.

βˆ’218βˆ’48+427+4128=βˆ’2(32)βˆ’4(22)+4(33)+4(82)-2 \sqrt{18} - 4 \sqrt{8} + 4 \sqrt{27} + 4 \sqrt{128} = -2(3 \sqrt{2}) - 4(2 \sqrt{2}) + 4(3 \sqrt{3}) + 4(8 \sqrt{2})

=βˆ’62βˆ’82+123+322= -6 \sqrt{2} - 8 \sqrt{2} + 12 \sqrt{3} + 32 \sqrt{2}

=βˆ’62βˆ’82+123+322= -6 \sqrt{2} - 8 \sqrt{2} + 12 \sqrt{3} + 32 \sqrt{2}

=βˆ’222+123= -22 \sqrt{2} + 12 \sqrt{3}

Conclusion

In this article, we simplified the given expression: βˆ’218βˆ’48+427+4128-2 \sqrt{18} - 4 \sqrt{8} + 4 \sqrt{27} + 4 \sqrt{128}. We broke down the expression into manageable parts, simplified each radical, and then combined the results to obtain the final simplified expression. By following these steps, we can simplify any radical expression and reveal its underlying structure.

Frequently Asked Questions

Q: What is a radical expression?

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

Q: How do I simplify a radical expression?

A: To simplify a radical expression, factor out perfect squares from under the radical sign.

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 combine simplified radicals?

A: To combine simplified radicals, add or subtract the coefficients of the radicals, and keep the same radical.

Final Answer

The final simplified expression is: βˆ’222+123-22 \sqrt{2} + 12 \sqrt{3}

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Introduction

Radical expressions can be complex and intimidating, but with the right techniques, they can be simplified to reveal their underlying structure. In this article, we will explore the process of simplifying radical expressions, focusing on the given expression: βˆ’218βˆ’48+427+4128-2 \sqrt{18} - 4 \sqrt{8} + 4 \sqrt{27} + 4 \sqrt{128}. We will break down the expression into manageable parts, simplify each radical, and then combine the results to obtain the final simplified expression.

Q&A: Simplifying Radical Expressions

Q: What is a radical expression?

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

Q: How do I simplify a radical expression?

A: To simplify a radical expression, factor out perfect squares from under the radical sign.

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 combine simplified radicals?

A: To combine simplified radicals, add or subtract the coefficients of the radicals, and keep the same radical.

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

A: A radical expression is an expression that contains a square root or other root of a number, while a rational expression is an expression that contains a fraction.

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

A: To simplify a radical expression with multiple terms, simplify each term separately and then combine the results.

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

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

  1. Simplify the terms inside the radical sign.
  2. Factor out perfect squares from under the radical sign.
  3. Simplify the resulting expression.

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

A: To simplify a radical expression with a negative coefficient, multiply the coefficient by -1 and then simplify the resulting expression.

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

A: A radical expression is an expression that contains a square root or other root of a number, while an exponent is a power to which a number is raised.

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

A: To simplify a radical expression with a variable, factor out the variable from under the radical sign and then simplify the resulting expression.

Examples of Simplifying Radical Expressions

Example 1: Simplify 16\sqrt{16}

16=4Γ—4=4Γ—4=4\sqrt{16} = \sqrt{4 \times 4} = \sqrt{4} \times \sqrt{4} = 4

Example 2: Simplify 9\sqrt{9}

9=3Γ—3=3Γ—3=3\sqrt{9} = \sqrt{3 \times 3} = \sqrt{3} \times \sqrt{3} = 3

Example 3: Simplify 25\sqrt{25}

25=5Γ—5=5Γ—5=5\sqrt{25} = \sqrt{5 \times 5} = \sqrt{5} \times \sqrt{5} = 5

Conclusion

In this article, we explored the process of simplifying radical expressions, focusing on the given expression: βˆ’218βˆ’48+427+4128-2 \sqrt{18} - 4 \sqrt{8} + 4 \sqrt{27} + 4 \sqrt{128}. We broke down the expression into manageable parts, simplified each radical, and then combined the results to obtain the final simplified expression. By following these steps, we can simplify any radical expression and reveal its underlying structure.

Frequently Asked Questions

Q: What is a radical expression?

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

Q: How do I simplify a radical expression?

A: To simplify a radical expression, factor out perfect squares from under the radical sign.

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 combine simplified radicals?

A: To combine simplified radicals, add or subtract the coefficients of the radicals, and keep the same radical.

Final Answer

The final simplified expression is: βˆ’222+123-22 \sqrt{2} + 12 \sqrt{3}