Simplify The Following Expressions Using The Imaginary Number.1. $\sqrt{-5}$2. $\sqrt{-49}$3. $-3 \sqrt{-9}$4. $-8 \sqrt{-64}$5. $-\sqrt{-50}$6. $-2 \sqrt{-12}$7. $5\sqrt{-12}$8.

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Introduction

Imaginary numbers are a fundamental concept in mathematics, particularly in algebra and calculus. They are used to extend the real number system to the complex number system, which includes all numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1. In this article, we will simplify expressions using imaginary numbers.

Simplifying Expressions

1. −5\sqrt{-5}

To simplify −5\sqrt{-5}, we can rewrite it as −1⋅5\sqrt{-1} \cdot \sqrt{5}. Since −1=i\sqrt{-1} = i, we have:

−5=i5\sqrt{-5} = i\sqrt{5}

2. −49\sqrt{-49}

To simplify −49\sqrt{-49}, we can rewrite it as −1⋅49\sqrt{-1} \cdot \sqrt{49}. Since −1=i\sqrt{-1} = i and 49=7\sqrt{49} = 7, we have:

−49=i⋅7=7i\sqrt{-49} = i \cdot 7 = 7i

3. −3−9-3 \sqrt{-9}

To simplify −3−9-3 \sqrt{-9}, we can rewrite it as −3⋅−1⋅9-3 \cdot \sqrt{-1} \cdot \sqrt{9}. Since −1=i\sqrt{-1} = i and 9=3\sqrt{9} = 3, we have:

−3−9=−3⋅i⋅3=−9i-3 \sqrt{-9} = -3 \cdot i \cdot 3 = -9i

4. −8−64-8 \sqrt{-64}

To simplify −8−64-8 \sqrt{-64}, we can rewrite it as −8⋅−1⋅64-8 \cdot \sqrt{-1} \cdot \sqrt{64}. Since −1=i\sqrt{-1} = i and 64=8\sqrt{64} = 8, we have:

−8−64=−8⋅i⋅8=−64i-8 \sqrt{-64} = -8 \cdot i \cdot 8 = -64i

5. −−50-\sqrt{-50}

To simplify −−50-\sqrt{-50}, we can rewrite it as −−1⋅50-\sqrt{-1} \cdot \sqrt{50}. Since −1=i\sqrt{-1} = i and 50=25⋅2=52\sqrt{50} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}, we have:

−−50=−i⋅52=−5i2-\sqrt{-50} = -i \cdot 5\sqrt{2} = -5i\sqrt{2}

6. −2−12-2 \sqrt{-12}

To simplify −2−12-2 \sqrt{-12}, we can rewrite it as −2⋅−1⋅12-2 \cdot \sqrt{-1} \cdot \sqrt{12}. Since −1=i\sqrt{-1} = i and 12=4⋅3=23\sqrt{12} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}, we have:

−2−12=−2⋅i⋅23=−4i3-2 \sqrt{-12} = -2 \cdot i \cdot 2\sqrt{3} = -4i\sqrt{3}

7. 5−125\sqrt{-12}

To simplify 5−125\sqrt{-12}, we can rewrite it as 5⋅−1⋅125 \cdot \sqrt{-1} \cdot \sqrt{12}. Since −1=i\sqrt{-1} = i and 12=4⋅3=23\sqrt{12} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}, we have:

5−12=5⋅i⋅23=10i35\sqrt{-12} = 5 \cdot i \cdot 2\sqrt{3} = 10i\sqrt{3}

Conclusion

In this article, we simplified expressions using imaginary numbers. We used the properties of imaginary numbers, particularly the fact that −1=i\sqrt{-1} = i, to rewrite each expression in a simpler form. The simplified expressions are:

  • −5=i5\sqrt{-5} = i\sqrt{5}
  • −49=7i\sqrt{-49} = 7i
  • −3−9=−9i-3 \sqrt{-9} = -9i
  • −8−64=−64i-8 \sqrt{-64} = -64i
  • −−50=−5i2-\sqrt{-50} = -5i\sqrt{2}
  • −2−12=−4i3-2 \sqrt{-12} = -4i\sqrt{3}
  • 5−12=10i35\sqrt{-12} = 10i\sqrt{3}

Introduction

Imaginary numbers are a fundamental concept in mathematics, particularly in algebra and calculus. They are used to extend the real number system to the complex number system, which includes all numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as the square root of -1. In this article, we will answer some frequently asked questions about imaginary numbers.

Q&A

Q: What is an imaginary number?

A: An imaginary number is a number that can be expressed as the product of a real number and the imaginary unit i, where i is defined as the square root of -1.

Q: What is the imaginary unit i?

A: The imaginary unit i is a number that satisfies the equation i^2 = -1. It is used to extend the real number system to the complex number system.

Q: How do I simplify expressions with imaginary numbers?

A: To simplify expressions with imaginary numbers, you can use the properties of imaginary numbers, particularly the fact that i^2 = -1. You can also use the fact that the square root of a negative number can be rewritten as the product of the square root of the absolute value of the number and the imaginary unit i.

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

A: A real number is a number that can be expressed as a finite decimal or fraction, such as 3 or 1/2. An imaginary number, on the other hand, is a number that can be expressed as the product of a real number and the imaginary unit i.

Q: Can imaginary numbers be used in real-world applications?

A: Yes, imaginary numbers have many real-world applications, including electrical engineering, signal processing, and quantum mechanics.

Q: How do I add and subtract imaginary numbers?

A: To add and subtract imaginary numbers, you can use the same rules as for real numbers, but you must also consider the imaginary unit i. For example, (3 + 4i) + (2 - 3i) = (3 + 2) + (4i - 3i) = 5 + i.

Q: How do I multiply and divide imaginary numbers?

A: To multiply and divide imaginary numbers, you can use the same rules as for real numbers, but you must also consider the imaginary unit i. For example, (3 + 4i) * (2 - 3i) = (3 * 2) + (3 * -3i) + (4i * 2) + (4i * -3i) = 6 - 9i + 8i - 12i^2 = 6 - i + 12 = 18 - i.

Q: What is the conjugate of an imaginary number?

A: The conjugate of an imaginary number is the number with the opposite sign of the imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.

Q: How do I find the square root of an imaginary number?

A: To find the square root of an imaginary number, you can use the fact that the square root of a negative number can be rewritten as the product of the square root of the absolute value of the number and the imaginary unit i.

Conclusion

In this article, we answered some frequently asked questions about imaginary numbers. We covered topics such as the definition of imaginary numbers, simplifying expressions with imaginary numbers, and real-world applications of imaginary numbers. We also covered topics such as adding and subtracting imaginary numbers, multiplying and dividing imaginary numbers, and finding the conjugate and square root of an imaginary number.