Write Each Expression As A Complex Number In Standard Form.a. − 25 − − 9 + − 81 \sqrt{-25} - \sqrt{-9} + \sqrt{-81} − 25 ​ − − 9 ​ + − 81 ​ B. − 27 + − 49 − − 64 \sqrt{-27} + \sqrt{-49} - \sqrt{-64} − 27 ​ + − 49 ​ − − 64 ​

Introduction

In mathematics, complex numbers are a fundamental concept that extends the real number system to include numbers with both real and imaginary parts. The standard form of a complex number is given by $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, which satisfies $i^2 = -1$. In this article, we will focus on rewriting given expressions as complex numbers in standard form.

Rewrite Each Expression as a Complex Number in Standard Form

a. $\sqrt{-25} - \sqrt{-9} + \sqrt{-81}$

To rewrite this expression as a complex number in standard form, we need to simplify each square root term. We can do this by factoring out the negative sign and expressing each term as a product of a real number and $i$.

$\sqrt{-25} = \sqrt{25 \cdot (-1)} = 5\sqrt{-1} = 5i$

$\sqrt{-9} = \sqrt{9 \cdot (-1)} = 3\sqrt{-1} = 3i$

$\sqrt{-81} = \sqrt{81 \cdot (-1)} = 9\sqrt{-1} = 9i$

Now, we can substitute these simplified expressions back into the original expression:

$\sqrt{-25} - \sqrt{-9} + \sqrt{-81} = 5i - 3i + 9i = 11i$

Therefore, the expression $\sqrt{-25} - \sqrt{-9} + \sqrt{-81}$ can be rewritten as the complex number $11i$ in standard form.

b. $\sqrt{-27} + \sqrt{-49} - \sqrt{-64}$

To rewrite this expression as a complex number in standard form, we need to simplify each square root term. We can do this by factoring out the negative sign and expressing each term as a product of a real number and $i$.

$\sqrt{-27} = \sqrt{27 \cdot (-1)} = 3\sqrt{3} \cdot \sqrt{-1} = 3\sqrt{3}i$

$\sqrt{-49} = \sqrt{49 \cdot (-1)} = 7\sqrt{-1} = 7i$

$\sqrt{-64} = \sqrt{64 \cdot (-1)} = 8\sqrt{-1} = 8i$

Now, we can substitute these simplified expressions back into the original expression:

$\sqrt{-27} + \sqrt{-49} - \sqrt{-64} = 3\sqrt{3}i + 7i - 8i = 3\sqrt{3}i - i$

Therefore, the expression $\sqrt{-27} + \sqrt{-49} - \sqrt{-64}$ can be rewritten as the complex number $3\sqrt{3}i - i$ in standard form.

Conclusion

In this article, we have rewritten two given expressions as complex numbers in standard form. We have simplified each square root term by factoring out the negative sign and expressing each term as a product of a real number and $i$. The resulting complex numbers are $11i$ and $3\sqrt{3}i - i$. These results demonstrate the importance of complex numbers in mathematics and their ability to represent a wide range of mathematical expressions.

Applications of Complex Numbers

Complex numbers have numerous applications in mathematics and other fields. Some of the key applications include:

• Algebra: Complex numbers are used to solve polynomial equations and to find the roots of quadratic equations.
• Geometry: Complex numbers are used to represent points in the complex plane and to perform geometric transformations.
• Trigonometry: Complex numbers are used to represent trigonometric functions and to solve trigonometric equations.
• Physics: Complex numbers are used to represent physical quantities such as voltage and current in electrical circuits.
• Engineering: Complex numbers are used to represent physical quantities such as impedance and admittance in electrical circuits.

Future Directions

Complex numbers are a fundamental concept in mathematics and have numerous applications in other fields. As mathematics continues to evolve, it is likely that complex numbers will play an increasingly important role in solving mathematical problems and modeling real-world phenomena.

References

• "Complex Numbers" by Math Open Reference. Retrieved from https://www.mathopenref.com/complexnumbers.html
• "Complex Numbers in Standard Form" by Khan Academy. Retrieved from https://www.khanacademy.org/math/pre-algebra/pre-algebra-review/complex-numbers/v/complex-numbers-in-standard-form

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of sources on the topic of complex numbers.