Which of the following is equivalent to 2i 6 7i? This question delves into the fascinating world of algebraic expressions, where simplifying and finding equivalent forms play a crucial role. By embarking on this mathematical journey, we will uncover the techniques and applications of this fundamental concept, unraveling the secrets that lie within algebraic expressions.
Simplifying expressions involves breaking them down into their simplest components, while finding equivalent expressions means discovering different representations that hold the same mathematical value. This exploration will shed light on these processes, providing a deeper understanding of algebraic manipulations and their significance in various fields.
Equivalent Expressions of 2i 6 7i
The given expression “2i 6 7i” is a complex number in the form a + bi, where a and b are real numbers and i is the imaginary unit. In this case, a = 6 and b = 9, so the expression can be rewritten as 6 + 9i.
The objective of this analysis is to find equivalent expressions for the given expression. Equivalent expressions are expressions that have the same value for all values of the variables involved.
Simplifying the Expression
Simplifying an algebraic expression means transforming it into an equivalent expression that is simpler or easier to work with. To simplify the expression “2i 6 7i”, we can use the following mathematical rules:
- Distributive property: a(b + c) = ab + ac
- Commutative property: a + b = b + a
- Associative property: (a + b) + c = a + (b + c)
Using these rules, we can simplify the expression as follows:
2i 6 7i = 2i + 6 + 7i = 6 + 9i
Identifying Equivalent Expressions
Two expressions are equivalent if they have the same value for all values of the variables involved. There are different methods for finding equivalent expressions, including:
- Factoring
- Expanding
- Using algebraic identities
In this case, we can use the following algebraic identity to find an equivalent expression for “6 + 9i”:
a + bi = r(cos θ + i sin θ)
where r is the modulus of the complex number and θ is its argument.
Exploring Equivalent Forms, Which of the following is equivalent to 2i 6 7i
Using the algebraic identity above, we can find various equivalent forms of “6 + 9i”. The following table shows some of these equivalent forms:
| Equivalent Form | Explanation |
|---|---|
| 6 + 9i | Original expression |
| 9(cos θ + i sin θ) | Using r = 9 and θ = tan-1(9/6) |
| 9(0.6 + 0.8i) | Approximating θ using a calculator |
Applications and Examples
Finding equivalent expressions is useful in various practical applications, such as:
- Solving complex equations
- Simplifying complex expressions
- Finding the modulus and argument of a complex number
For example, in electrical engineering, equivalent expressions are used to simplify circuit analysis and design.
Question & Answer Hub: Which Of The Following Is Equivalent To 2i 6 7i
What is the purpose of finding equivalent expressions?
Finding equivalent expressions allows us to simplify complex expressions, solve equations more efficiently, and represent mathematical concepts in different ways.
How can we determine if two expressions are equivalent?
Expressions are equivalent if they have the same value for all possible values of the variables involved.
What are some common methods for finding equivalent expressions?
Common methods include factoring, expanding, using algebraic identities, and applying mathematical operations.
