Problem 51 Multiply. $$ (4+5 i)^{2} $... [FREE SOLUTION]

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Chapter 8: Problem 51

Multiply. $$ (4+5 i)^{2} $$

Short Answer

Expert verified

-9 + 40i

Step by step solution

Write the expression

Start with the given expression: (4 + 5i)^{2}

Apply the binomial theorem

Use the binomial theorem to expand the expression: (a + b)^{2} = a^{2} + 2ab + b^{2}. Here, a = 4 and b = 5i.

Calculate each term

Calculate each term in the expansion: 4^{2} = 16 2(4)(5i) = 40i (5i)^{2} = 25i^{2} Recall that i^{2} = 鈭�1, so 25i^{2} = 25(鈭�1) = 鈭�25.

Sum the terms

Combine the calculated terms to get the final result: 16 + 40i - 25.

Simplify the expression

Combine the real parts: 16 - 25 = -9. Thus, the simplified form is: -9 + 40i.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Theorem

The binomial theorem allows us to expand expressions of the form $(a + b)^n$, where 'a' and 'b' are any numbers, and 'n' is a positive integer. For example,
(4 + 5i)^2.
When n = 2, the binomial theorem states:
(a + b)^2 = a^2 + 2ab + b^2.
In the given exercise,
a = 4 and
b = 5i.
Substituting these values into the binomial expansion,
(4 + 5i)^2 = 4^2 + 2(4)(5i) + (5i)^2.
This expansion simplifies the multiplication by breaking it down into manageable steps. For a^2 = 16,
2ab = 40i,
and
b^2 = 25i^2.

Complex Numbers

Complex numbers are expressed as $a + bi$,
where 'a' and 'b' are real numbers, and $i$ is the imaginary unit with the property that $i^2 = -1$.
In our exercise, 4 + 5i is a complex number with the real part 4 and the imaginary part 5.
When performing mathematical operations on complex numbers, ensure you keep the real and imaginary parts distinct.
Recall the important identity:
i^2 = -1.
This identity is crucial when squaring or multiplying complex numbers. For example:
(5i)^2
is calculated as:
25i^2 = 25(-1) = -25. This helps to convert complex imaginaries back to real numbers.

Algebraic Expansion

Algebraic expansion involves multiplying out the terms in an expression. For example,
(4 + 5i)^2
expands into three terms:
4^2, 2(4)(5i), and (5i)^2.
Once expanded:
4^2 = 16,
2(4)(5i) = 40i, and
25i^2 = -25. Next, sum these terms:
16 + 40i - 25.This result contains both a real part (16 - 25) and an imaginary part (40i). Finally,
combine the real parts: 16 - 25 = -9., and
combine with the imaginary part to get $-9 + 40i$.
This step-by-step simplification ensures we correctly multiply complex numbers and expand algebraic terms.

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91影视

Multiply. $$ (4+5 i)^{2} $$

Short Answer

Step by step solution

Write the expression

Apply the binomial theorem

Calculate each term

Sum the terms

Simplify the expression

Key Concepts

Binomial Theorem

Complex Numbers

Algebraic Expansion

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