Simplify The Expression:$ -3 \sqrt{27} + 2 \sqrt{12} $

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Introduction

Simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. In this article, we will focus on simplifying the given expression: $-3 \sqrt{27} + 2 \sqrt{12}$. We will break down the process step by step, using various mathematical techniques to simplify the expression.

Understanding Square Roots

Before we dive into simplifying the expression, let's quickly review what square roots are. A 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. We denote the square root of a number using the symbol $\sqrt{}$. For instance, $\sqrt{16}$ represents the square root of 16.

Simplifying $\sqrt{27}$

The first step in simplifying the expression is to simplify $\sqrt{27}$. To do this, we need to find the largest perfect square that divides 27. A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as $4^2$. Similarly, 9 is a perfect square because it can be expressed as $3^2$.

We can rewrite 27 as $9 \times 3$. Since 9 is a perfect square, we can simplify $\sqrt{27}$ as follows:

$$\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}$$

Simplifying $\sqrt{12}$

Next, we need to simplify $\sqrt{12}$. To do this, we can rewrite 12 as $4 \times 3$. Since 4 is a perfect square, we can simplify $\sqrt{12}$ as follows:

$$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$$

Substituting Simplified Values

Now that we have simplified $\sqrt{27}$ and $\sqrt{12}$, we can substitute these values back into the original expression:

$$-3 \sqrt{27} + 2 \sqrt{12} = -3(3\sqrt{3}) + 2(2\sqrt{3})$$

Distributing Negative and Positive Signs

Next, we need to distribute the negative and positive signs to the terms inside the parentheses:

$$-3(3\sqrt{3}) + 2(2\sqrt{3}) = -9\sqrt{3} + 4\sqrt{3}$$

Combining Like Terms

Finally, we can combine the like terms:

$$-9\sqrt{3} + 4\sqrt{3} = -5\sqrt{3}$$

Conclusion

In this article, we simplified the expression $-3 \sqrt{27} + 2 \sqrt{12}$ using various mathematical techniques. We started by simplifying $\sqrt{27}$ and $\sqrt{12}$, and then substituted these values back into the original expression. We distributed the negative and positive signs to the terms inside the parentheses, and finally combined the like terms to get the simplified expression: $-5\sqrt{3}$.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions involving square roots:

• Look for perfect squares: When simplifying square roots, look for perfect squares that divide the number inside the square root.
• Use the product rule: When simplifying square roots, use the product rule to break down the number inside the square root into smaller factors.
• Distribute negative and positive signs: When simplifying expressions involving square roots, distribute the negative and positive signs to the terms inside the parentheses.
• Combine like terms: Finally, combine the like terms to get the simplified expression.

Frequently Asked Questions

Here are some frequently asked questions about simplifying expressions involving square roots:

• What is the square root of a number?: The square root of a number is a value that, when multiplied by itself, gives the original number.
• How do I simplify square roots?: To simplify square roots, look for perfect squares that divide the number inside the square root, use the product rule to break down the number inside the square root into smaller factors, distribute the negative and positive signs to the terms inside the parentheses, and finally combine the like terms.
• What is the difference between a perfect square and a square root?: A perfect square is a number that can be expressed as the square of an integer, while a square root is a value that, when multiplied by itself, gives the original number.

Final Thoughts

Simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, you can simplify expressions involving square roots and get the final answer. Remember to look for perfect squares, use the product rule, distribute negative and positive signs, and combine like terms to get the simplified expression.

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Introduction

In our previous article, we simplified the expression $-3 \sqrt{27} + 2 \sqrt{12}$ using various mathematical techniques. We started by simplifying $\sqrt{27}$ and $\sqrt{12}$, and then substituted these values back into the original expression. We distributed the negative and positive signs to the terms inside the parentheses, and finally combined the like terms to get the simplified expression: $-5\sqrt{3}$.

In this article, we will answer some frequently asked questions about simplifying expressions involving square roots. We will cover topics such as what is the square root of a number, how to simplify square roots, and the difference between a perfect square and a square root.

Q&A

Q: What is the square root of a number?

A: 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.

Q: How do I simplify square roots?

A: To simplify square roots, look for perfect squares that divide the number inside the square root, use the product rule to break down the number inside the square root into smaller factors, distribute the negative and positive signs to the terms inside the parentheses, and finally combine the like terms.

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

A: A perfect square is a number that can be expressed as the square of an integer, while a square root is a value that, when multiplied by itself, gives the original number. For example, 16 is a perfect square because it can be expressed as $4^2$, while $\sqrt{16}$ is a square root because it is a value that, when multiplied by itself, gives 16.

Q: How do I know if a number is a perfect square?

A: To determine if a number is a perfect square, look for a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as $4^2$, while 27 is not a perfect square because it cannot be expressed as the square of an integer.

Q: Can I simplify square roots with variables?

A: Yes, you can simplify square roots with variables. To do this, look for perfect squares that divide the variable inside the square root, use the product rule to break down the variable inside the square root into smaller factors, distribute the negative and positive signs to the terms inside the parentheses, and finally combine the like terms.

Q: How do I simplify square roots with fractions?

A: To simplify square roots with fractions, look for perfect squares that divide the numerator and denominator of the fraction, use the product rule to break down the numerator and denominator of the fraction into smaller factors, distribute the negative and positive signs to the terms inside the parentheses, and finally combine the like terms.

Q: Can I simplify square roots with decimals?

A: Yes, you can simplify square roots with decimals. To do this, look for perfect squares that divide the decimal inside the square root, use the product rule to break down the decimal inside the square root into smaller factors, distribute the negative and positive signs to the terms inside the parentheses, and finally combine the like terms.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions involving square roots:

• Look for perfect squares: When simplifying square roots, look for perfect squares that divide the number inside the square root.
• Use the product rule: When simplifying square roots, use the product rule to break down the number inside the square root into smaller factors.
• Distribute negative and positive signs: When simplifying expressions involving square roots, distribute the negative and positive signs to the terms inside the parentheses.
• Combine like terms: Finally, combine the like terms to get the simplified expression.

Frequently Asked Questions

Here are some frequently asked questions about simplifying expressions involving square roots:

• What is the square root of a number?: The square root of a number is a value that, when multiplied by itself, gives the original number.
• How do I simplify square roots?: To simplify square roots, look for perfect squares that divide the number inside the square root, use the product rule to break down the number inside the square root into smaller factors, distribute the negative and positive signs to the terms inside the parentheses, and finally combine the like terms.
• What is the difference between a perfect square and a square root?: A perfect square is a number that can be expressed as the square of an integer, while a square root is a value that, when multiplied by itself, gives the original number.

Final Thoughts

Simplifying expressions involving square roots is a crucial skill in mathematics, particularly in algebra and geometry. By following the steps outlined in this article, you can simplify expressions involving square roots and get the final answer. Remember to look for perfect squares, use the product rule, distribute negative and positive signs, and combine like terms to get the simplified expression.