Evaluate The Expression: ${ 7 \sqrt{-162} }$

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Introduction

When evaluating mathematical expressions, it's essential to understand the rules and properties of the operations involved. In this case, we're dealing with a square root of a negative number, which can be a bit tricky. The expression $\sqrt{-162}$ requires us to apply the properties of square roots and complex numbers. In this article, we'll break down the steps to evaluate this expression and provide a clear understanding of the concepts involved.

Understanding Square Roots of Negative Numbers

The square root of a negative number is a complex number, which can be represented in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as the square root of $-1$. When we encounter a square root of a negative number, we need to express it in terms of complex numbers.

Simplifying the Expression

To simplify the expression $\sqrt{-162}$, we can start by factoring the number under the square root sign. We can write $-162$ as $-1 \times 162$, which can be further factored as $-1 \times 2 \times 81$. This gives us:

$$\sqrt{-162} = \sqrt{-1 \times 2 \times 81}$$

Applying the Properties of Square Roots

Now, we can apply the properties of square roots to simplify the expression further. We know that the square root of a product is equal to the product of the square roots. Therefore, we can write:

$$\sqrt{-1 \times 2 \times 81} = \sqrt{-1} \times \sqrt{2} \times \sqrt{81}$$

Evaluating the Square Roots

Next, we need to evaluate the square roots of the individual numbers. We know that $\sqrt{-1} = i$, $\sqrt{2}$ is a real number, and $\sqrt{81} = 9$. Therefore, we can substitute these values into the expression:

$$\sqrt{-1} \times \sqrt{2} \times \sqrt{81} = i \times \sqrt{2} \times 9$$

Simplifying the Expression Further

Now, we can simplify the expression further by multiplying the numbers together:

$$i \times \sqrt{2} \times 9 = 9i\sqrt{2}$$

Conclusion

In conclusion, the expression $\sqrt{-162}$ can be evaluated as $9i\sqrt{2}$. This involves applying the properties of square roots and complex numbers to simplify the expression. We hope this article has provided a clear understanding of the concepts involved and has helped you evaluate similar expressions in the future.

Frequently Asked Questions

• What is the square root of a negative number?
• How do you simplify the expression $\sqrt{-162}$?
• What is the value of $i$ in the expression $9i\sqrt{2}$?

Answers

• The square root of a negative number is a complex number, which can be represented in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as the square root of $-1$.
• To simplify the expression $\sqrt{-162}$, we can factor the number under the square root sign and apply the properties of square roots.
• The value of $i$ in the expression $9i\sqrt{2}$ is the imaginary unit, which is defined as the square root of $-1$.

Related Topics

• Complex Numbers
• Square Roots
• Imaginary Unit

Further Reading

• For more information on complex numbers, see the article "Understanding Complex Numbers".
• For more information on square roots, see the article "Evaluating Square Roots".
• For more information on the imaginary unit, see the article "Understanding the Imaginary Unit".
  

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Introduction

In our previous article, we evaluated the expression $\sqrt{-162}$ and found that it can be simplified to $9i\sqrt{2}$. However, we know that there are many more questions and doubts that readers may have about this topic. In this article, we'll address some of the most frequently asked questions and provide clear and concise answers.

Q&A

Q: What is the square root of a negative number?

A: The square root of a negative number is a complex number, which can be represented in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as the square root of $-1$.

Q: How do you simplify the expression $\sqrt{-162}$?

A: To simplify the expression $\sqrt{-162}$, we can factor the number under the square root sign and apply the properties of square roots. We can write $-162$ as $-1 \times 162$, which can be further factored as $-1 \times 2 \times 81$. This gives us:

$$\sqrt{-162} = \sqrt{-1 \times 2 \times 81}$$

We can then apply the properties of square roots to simplify the expression further:

$$\sqrt{-1 \times 2 \times 81} = \sqrt{-1} \times \sqrt{2} \times \sqrt{81}$$

Q: What is the value of $i$ in the expression $9i\sqrt{2}$?

A: The value of $i$ in the expression $9i\sqrt{2}$ is the imaginary unit, which is defined as the square root of $-1$. In other words, $i$ is a mathematical constant that represents the square root of $-1$.

Q: Can you provide more examples of simplifying expressions with square roots of negative numbers?

A: Yes, here are a few more examples:

• $\sqrt{-25} = \sqrt{-1 \times 25} = \sqrt{-1} \times \sqrt{25} = 5i$
• $\sqrt{-36} = \sqrt{-1 \times 36} = \sqrt{-1} \times \sqrt{36} = 6i$
• $\sqrt{-49} = \sqrt{-1 \times 49} = \sqrt{-1} \times \sqrt{49} = 7i$

Q: How do you add and subtract complex numbers?

A: To add and subtract complex numbers, we can use the following rules:

• To add two complex numbers, we add the real parts and the imaginary parts separately.
• To subtract two complex numbers, we subtract the real parts and the imaginary parts separately.

For example:

• $(3 + 4i) + (2 + 5i) = (3 + 2) + (4 + 5)i = 5 + 9i$
• $(3 + 4i) - (2 + 5i) = (3 - 2) + (4 - 5)i = 1 - i$

Q: Can you provide more information on complex numbers?

A: Yes, complex numbers are a fundamental concept in mathematics, and they have many applications in fields such as physics, engineering, and computer science. Complex numbers are numbers that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

Conclusion

In conclusion, we hope this Q&A article has provided a clear and concise understanding of the concepts involved in evaluating the expression $\sqrt{-162}$. We've addressed some of the most frequently asked questions and provided examples and explanations to help readers understand the material. If you have any further questions or doubts, please don't hesitate to ask.

Related Topics

• Complex Numbers
• Square Roots
• Imaginary Unit

Further Reading

• For more information on complex numbers, see the article "Understanding Complex Numbers".
• For more information on square roots, see the article "Evaluating Square Roots".
• For more information on the imaginary unit, see the article "Understanding the Imaginary Unit".