Which Of The Following Is Equivalent To The Expression Below? − 300 \sqrt{-300} − 300 ​ A. − 10 I 3 -10 I \sqrt{3} − 10 I 3 ​ B. 3 I 10 3 I \sqrt{10} 3 I 10 ​ C. − 3 I 10 -3 I \sqrt{10} − 3 I 10 ​ D. 10 I 3 10 I \sqrt{3} 10 I 3 ​

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Introduction

When dealing with complex numbers, simplifying square roots can be a challenging task. In this article, we will explore the process of simplifying complex square roots and apply this knowledge to the given expression −300\sqrt{-300}. We will examine each option and determine which one is equivalent to the given expression.

Understanding Complex Numbers

Before we dive into simplifying the square root, let's briefly review complex numbers. A complex number is a number that can be expressed in the form a+bia + bi, where aa and bb are real numbers and ii is the imaginary unit, which satisfies the equation i2=−1i^2 = -1. The real part of a complex number is aa, and the imaginary part is bb.

Simplifying Square Roots

To simplify the square root of a negative number, we can use the following property:

−a=ia\sqrt{-a} = i\sqrt{a}

This property allows us to rewrite the square root of a negative number as the product of ii and the square root of the absolute value of the number.

Applying the Property to the Given Expression

Now, let's apply this property to the given expression −300\sqrt{-300}. We can rewrite it as:

−300=i300\sqrt{-300} = i\sqrt{300}

Next, we can simplify the square root of 300 by factoring it into prime factors:

300=100⋅3=103\sqrt{300} = \sqrt{100 \cdot 3} = 10\sqrt{3}

Therefore, the simplified expression is:

−300=i⋅103=10i3\sqrt{-300} = i \cdot 10\sqrt{3} = 10i\sqrt{3}

Comparing the Simplified Expression to the Options

Now that we have simplified the expression, let's compare it to the options:

A. −10i3-10i\sqrt{3} B. 3i103i\sqrt{10} C. −3i10-3i\sqrt{10} D. 10i310i\sqrt{3}

Based on our simplified expression, we can see that option D is the only one that matches our result.

Conclusion

In this article, we simplified the complex square root −300\sqrt{-300} using the property −a=ia\sqrt{-a} = i\sqrt{a}. We then compared our simplified expression to the given options and determined that option D, 10i310i\sqrt{3}, is the correct answer.

Key Takeaways

  • Simplifying complex square roots involves using the property −a=ia\sqrt{-a} = i\sqrt{a}.
  • Factoring the number under the square root into prime factors can help simplify the expression.
  • When comparing the simplified expression to the options, pay close attention to the signs and the values of the real and imaginary parts.

Additional Resources

For more information on complex numbers and simplifying square roots, check out the following resources:

  • Khan Academy: Complex Numbers
  • Math Is Fun: Complex Numbers
  • Wolfram MathWorld: Complex Numbers

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed without any imaginary part, such as 3 or -4. A complex number, on the other hand, is a number that can be expressed in the form a+bia + bi, where aa and bb are real numbers and ii is the imaginary unit, which satisfies the equation i2=−1i^2 = -1.

Q: How do I simplify a complex square root?

A: To simplify a complex square root, you can use the property −a=ia\sqrt{-a} = i\sqrt{a}. This property allows you to rewrite the square root of a negative number as the product of ii and the square root of the absolute value of the number.

Q: What is the absolute value of a complex number?

A: The absolute value of a complex number a+bia + bi is given by a2+b2\sqrt{a^2 + b^2}. This is also known as the magnitude or modulus of the complex number.

Q: How do I factor a number under a square root?

A: To factor a number under a square root, you can look for perfect squares that divide the number. For example, if you have 300\sqrt{300}, you can factor it as 100⋅3=103\sqrt{100 \cdot 3} = 10\sqrt{3}.

Q: What is the difference between ii and −i-i?

A: ii and −i-i are both imaginary units, but they have opposite signs. When you multiply a complex number by ii, you are essentially rotating it by 90 degrees counterclockwise. When you multiply it by −i-i, you are rotating it by 90 degrees clockwise.

Q: Can I simplify a complex square root with a negative number under the square root?

A: Yes, you can simplify a complex square root with a negative number under the square root using the property −a=ia\sqrt{-a} = i\sqrt{a}. For example, −300=i300=i⋅103=10i3\sqrt{-300} = i\sqrt{300} = i \cdot 10\sqrt{3} = 10i\sqrt{3}.

Q: How do I know which option is correct when simplifying a complex square root?

A: When simplifying a complex square root, you should pay close attention to the signs and the values of the real and imaginary parts. Make sure to use the correct property and factor the number under the square root correctly.

Q: What are some common mistakes to avoid when simplifying complex square roots?

A: Some common mistakes to avoid when simplifying complex square roots include:

  • Not using the correct property
  • Not factoring the number under the square root correctly
  • Not paying attention to the signs and the values of the real and imaginary parts
  • Not checking your work carefully

Q: Where can I find more resources on simplifying complex square roots?

A: There are many resources available online that can help you learn more about simplifying complex square roots, including:

  • Khan Academy: Complex Numbers
  • Math Is Fun: Complex Numbers
  • Wolfram MathWorld: Complex Numbers
  • Online math textbooks and tutorials

By following these tips and practicing with more examples, you will become more comfortable simplifying complex square roots and solving problems involving complex numbers.