Which expressions are equal to a real number?

Option B

Option C

Step-by-step explanation:

⇒ , which is a real number.

Similarly,

And we also know that

So, we see that odd powers of is a complex number and even powers renders us a real number.

Using this we solve the problem.

Option A: 11 is an odd power. This would be a complex number and not a real number.

Option B: Even power of i. So, this should give us a real number.

Option C:

                            (a + bi)² = a² - b² + 2abi

So, (2 + 3i)² would be a complex number because of the 2bi term.

Option D: (4 + 5i)(4 - 5i)

                         (a + ib)(a - ib) = a² + b²

So, (4 + 5i)(4 - 5i) = 4² + 5² = 41, a real number.

Option E: (6 + 8i)(8 + 6i)

                     (a + ib)(c + id) = ac - bd + (ad + bc)i

This would be a complex number because of the (ad + bc) term.

So, Options B and D are real numbers.

sup

step-by-step explanation:

∠1≅∠2 by the alternate exterior angles theorem.

step-by-step explanation:

given, a ∥ b and ∠1 ≅ ∠3 .we have to prove that e ∥ f

we know that ∠1≅∠3 and that a || b because they are given. we see that by the alternate exterior angles theorem. therefore, ∠2≅∠3 by the transitive property. so, we can conclude that e || f by the converse alternate exterior angles theorem.

we have to fill the missing statement.

transitivity property states that if a = b and b = c, then a = c.

now, given ∠1≅∠3 and by transitivity property ∠2≅∠3 .

hence, to apply transitivity property one angle must be common which is not in result after applying this property which is ∠1.

the only options in which ∠1 is present are ∠1 and ∠2, ∠1 and ∠4

∠1 and ∠4 is not possible ∵ after applying transitivity we didn't get ∠4.

hence, the missing statement is ∠1≅∠2.

so, ∠1≅∠2 by the alternate exterior angles theorem.