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Chapter 1: Problem 36

Evaluate the product, and write the result in the form \(a+b i\) $$(3-7 i)^{2}$$

Short Answer

Expert verified
The result is \(-40 - 42i\).

Step by step solution

01

Write the Expression

We need to evaluate the expression \((3-7i)^2\). This means we're squaring the complex number \(3 - 7i\).
02

Apply the Binomial Formula

The formula for squaring a binomial \((a - b)^2\) is \(a^2 - 2ab + b^2\). We identify that here, \(a = 3\) and \(b = 7i\).
03

Calculate Each Term

First calculate \(a^2 = 3^2 = 9\). Then calculate \(-2ab = -2 \times 3 \times 7i = -42i\). Lastly, calculate \(b^2 = (7i)^2 = 49i^2\). Recall that \(i^2 = -1\), thus \(49i^2 = 49 (-1) = -49\).
04

Combine the Terms

Combine all the parts: \(9 - 49 - 42i\). Simplify the real parts to get \(9 - 49 = -40\). Hence, the expression simplifies to \(-40 - 42i\).
05

Express the Final Result

The result of \((3-7i)^2\) is \(-40 - 42i\), which is in the form \(a+bi\) where \(a = -40\) and \(b = -42\).

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Key Concepts

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

Understanding the Binomial Formula
The binomial formula is a crucial mathematical tool used to expand expressions that are raised to a power. Specifically, when dealing with squaring a binomial, we use the formula \[(a - b)^2 = a^2 - 2ab + b^2\].This formula allows us to expand and simplify expressions like \((3-7i)^2\).
To apply it:
  • Identify the first term \(a\) and the second term \(b\) in the binomial.
  • Compute \(a^2\), which is squaring the first term.
  • Calculate \(-2ab\), which involves multiplying both terms by \(-2\).
  • Find \(b^2\), by squaring the second term.
By using this formula, it becomes straightforward to expand complex expressions and simplify them to solve problems, like squaring binomials with imaginary numbers.
The Imaginary Unit \(i\) and Its Square
The imaginary unit \(i\) is a fundamental concept in complex numbers, defined as the square root of \(-1\). This might seem strange, as there is no real number whose square is negative. Therefore, mathematicians invented \(i\).
The property \(i^2 = -1\) plays a critical role in simplifying expressions involving \(i\). When squaring a complex expression, remember to substitute \(i^2\) with \(-1\).
For example, in the expression \((3-7i)^2\), when calculating \(b^2\) where \(b = 7i\), the process is as follows:
  • Square \(7i\): \((7i)^2 = 49i^2\).
  • Substitute \(i^2\) with \(-1\) to get \(49(-1) = -49\).
This substitution is essential for correctly simplifying complex expressions, ensuring the real and imaginary parts are accurately determined.
Simplifying Algebraic Expressions
Algebraic expressions include both numbers and variables, such as the expression \((3-7i)^2\). Simplifying these expressions involves combining like terms and using algebraic properties, such as distribution and addition.
During simplification of a complex expression, follow these steps:
  • Distribute and apply mathematical operations as guided by formulas such as the binomial formula.
  • Combine like terms. For example, in \(9 - 49 - 42i\), you first combine \(9\) and \(-49\) to get \(-40\).
  • Ensure expressions result in the standard complex form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary coefficient.
Through these steps, complex and algebraic expressions become manageable, making it easier to interpret and solve mathematical problems.

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Most popular questions from this chapter

Prove the following Laws of Exponents. (a) Law \(6:\left(\frac{a}{b}\right)^{-n}=\frac{b^{n}}{a^{n}}\) (b) Law \(7: \frac{a^{-n}}{b^{-m}}=\frac{b^{m}}{a^{n}}\)

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt{4 s t^{3}} \sqrt[6]{s^{3} t^{2}}\) (b) \(\frac{\sqrt[4]{x^{7}}}{\sqrt[4]{x^{3}}}\)

Rational Exponents Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. (a) \(x^{3 / 4} x^{5 / 4}\) (b) \(y^{2 / 3} y^{4 / 3}\)

Manufacturer's Profit If a manufacturer sells \(x\) units of a certain product, revenue \(R\) and cost \(C\) (in dollars) are given by $$ \begin{array}{l} R=20 x \\ C=2000+8 x+0.0025 x^{2} \end{array} $$ Use the fact that profit \(=\) revenue \(-\) cost to determine how many units the manufacturer should sell to enjoy a profit of at least \(\$ 2400\).

Sketch the region given by the set. $$\left\\{(x, y) | x^{2}+y^{2}>4\right\\}$$
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