Use The Imaginary Number { I $}$ To Rewrite The Expression As A Complex Number: ${\sqrt{100} + \sqrt{-81}}$A. ${ 9i\$} B. ${ 10 + 9i\$} C. ${ 10 - 9i\$} D. ${ 19i\$} E. ${ 19i\$}
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. In this article, we will explore how to rewrite an expression as a complex number using the imaginary number i.
Understanding Imaginary Numbers
The imaginary unit i is defined as the square root of -1, denoted by i = √(-1). This means that i^2 = -1. Imaginary numbers are used to represent quantities that cannot be expressed on the real number line. They are used to solve equations that involve square roots of negative numbers.
Rewriting the Expression
The given expression is √100 + √(-81). To rewrite this expression as a complex number, we need to simplify the square roots.
Simplifying the Square Roots
The square root of 100 is 10, since 10^2 = 100. The square root of -81 is not a real number, but it can be expressed as a complex number. We can rewrite √(-81) as √(-1) × √81, using the property of square roots that √(ab) = √a × √b.
Applying the Property of Square Roots
√(-1) × √81 = i × 9, since i = √(-1) and 9^2 = 81.
Rewriting the Expression as a Complex Number
Now we can rewrite the original expression as 10 + i × 9. To simplify this expression, we can multiply i by 9, which gives us 9i.
Simplifying the Expression
10 + 9i is the simplified form of the original expression. This is a complex number, where 10 is the real part and 9i is the imaginary part.
Conclusion
In this article, we have seen how to rewrite an expression as a complex number using the imaginary number i. We have simplified the square roots and applied the property of square roots to rewrite the expression as a complex number. The final answer is 10 + 9i.
Answer Key
The correct answer is B. 10 + 9i.
Additional Examples
Here are some additional examples of rewriting expressions as complex numbers:
- √(-16) + √100 = 4i + 10
- √(-25) + √(-36) = 5i - 6i
- √(-49) + √(-64) = 7i - 8i
These examples demonstrate how to rewrite expressions as complex numbers using the imaginary number i.
Tips and Tricks
Here are some tips and tricks for working with imaginary numbers:
- Remember that i^2 = -1.
- Use the property of square roots that √(ab) = √a × √b.
- Simplify expressions by combining like terms.
- Use the distributive property to multiply complex numbers.
By following these tips and tricks, you can become proficient in working with imaginary numbers and rewriting expressions as complex numbers.
Common Mistakes
Here are some common mistakes to avoid when working with imaginary numbers:
- Forgetting to simplify expressions.
- Not using the property of square roots.
- Not remembering that i^2 = -1.
- Not using the distributive property to multiply complex numbers.
By avoiding these common mistakes, you can ensure that your work with imaginary numbers is accurate and correct.
Conclusion
Q: What is an imaginary number?
A: An imaginary number is a complex number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. The imaginary unit i is defined as the square root of -1, denoted by i = √(-1).
Q: What is the difference between a real number and an imaginary number?
A: A real number is a number that can be expressed on the real number line, whereas an imaginary number is a number that cannot be expressed on the real number line. Imaginary numbers are used to extend the real number system to the complex number system.
Q: How do I simplify an expression with imaginary numbers?
A: To simplify an expression with imaginary numbers, you need to follow the order of operations (PEMDAS):
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: What is the property of square roots that I need to remember?
A: The property of square roots that you need to remember is that √(ab) = √a × √b. This property allows you to simplify expressions with square roots.
Q: How do I multiply complex numbers?
A: To multiply complex numbers, you need to follow the distributive property:
(a + bi) × (c + di) = (ac - bd) + (ad + bc)i
Q: What is the difference between i and -i?
A: i and -i are conjugates of each other. They are used to simplify expressions with imaginary numbers. When you multiply a complex number by its conjugate, you get a real number.
Q: How do I add and subtract complex numbers?
A: To add and subtract complex numbers, you need to follow the rules of addition and subtraction:
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) - (c + di) = (a - c) + (b - d)i
Q: What is the significance of the imaginary unit i?
A: The imaginary unit i is a fundamental concept in mathematics. It is 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.
Q: How do I simplify an expression with multiple imaginary numbers?
A: To simplify an expression with multiple imaginary numbers, you need to follow the order of operations (PEMDAS) and use the properties of imaginary numbers. You can also use the distributive property to multiply complex numbers.
Q: What are some common mistakes to avoid when working with imaginary numbers?
A: Some common mistakes to avoid when working with imaginary numbers include:
- Forgetting to simplify expressions.
- Not using the property of square roots.
- Not remembering that i^2 = -1.
- Not using the distributive property to multiply complex numbers.
By avoiding these common mistakes, you can ensure that your work with imaginary numbers is accurate and correct.
Conclusion
In conclusion, imaginary numbers are a fundamental concept in mathematics. By understanding the properties of imaginary numbers and applying them to simplify expressions, you can become proficient in working with complex numbers. Remember to simplify expressions, use the property of square roots, and remember that i^2 = -1. With practice and patience, you can master the art of working with imaginary numbers.