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Chapter 12: Problem 19

Simplify each of the given expressions. (a) \(\sqrt{(-2)(-8)}\) (b) \(\sqrt{-2} \sqrt{-8}\)

Short Answer

Expert verified
(a) 4 and (b) -4.

Step by step solution

01

Simplify the Inside of the Square Root for (a)

In the expression \( \sqrt{(-2)(-8)} \), first simplify the multiplication inside the square root. Here, \((-2) \times (-8) = 16\). Thus, the expression simplifies to \( \sqrt{16} \).
02

Find the Square Root for (a)

Now, find the square root of the simplified number. \( \sqrt{16} = 4 \). Therefore, the simplified form of the expression is \( 4 \).
03

Simplify Each Square Root Factor for (b)

For \( \sqrt{-2} \sqrt{-8} \), recognize that each term under the square root is negative. Rewrite each square root in terms of \( i \), where \( i = \sqrt{-1} \). Thus, \( \sqrt{-2} = i \sqrt{2} \) and \( \sqrt{-8} = i \sqrt{8} \).
04

Simplify and Combine the Result for (b)

Multiply the results from Step 3: \( (i \sqrt{2})(i \sqrt{8}) = i^2 \times \sqrt{2 \times 8} = -1 \times \sqrt{16} \). Simplify \( \sqrt{16} = 4 \), leading to \( -1 \times 4 = -4 \).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is a fundamental concept in algebra that helps us make complex problems easier to handle. At its core, simplifying involves rewriting an expression in a way that is more straightforward or convenient. Often, this means condensing multiple terms into a single term.
  • Start by consolidating any like terms. This means combining terms that have the same variable parts.
  • When dealing with operations like multiplication and addition, follow the order of operations: parentheses, exponents, multiplication & division, and addition & subtraction (PEMDAS).
  • If you encounter complex expressions within parentheses, simplify the inside first.
By breaking down and rearranging expressions, students can find more intuitive solutions and reduce potential errors. Practice is key to mastering this skill, so simplify as often as possible to become more efficient.
Square Roots
Understanding square roots is essential in both algebra and higher-level mathematics. A square root of a number is a value that, when multiplied by itself, gives the original number. This operation is denoted by the symbol \( \sqrt{} \).
  • Basic square roots include numbers like \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \).
  • Finding the square root of perfect squares, like 16 or 25, is straightforward since they result in whole numbers.
  • When dealing with negative numbers, like \( \sqrt{-1} \), introduce complex numbers. This is where \( i \), the imaginary unit, becomes useful.
Square roots are crucial when simplifying expressions and solving quadratic equations. By understanding and applying the properties of square roots, students can simplify algebraic expressions more effectively.
Complex Numbers
Complex numbers expand the concept of numbers used in calculations and include a real part and an imaginary part. This allows for solutions to equations that do not have real-number solutions.
  • The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), meaning \( i^2 = -1 \).
  • Complex numbers are presented as \( a + bi \), where \( a \) and \( b \) are real numbers.
  • Addition and subtraction of complex numbers involve combining like terms (real with real, imaginary with imaginary).
In the exercise provided, complex numbers arise when simplifying square roots of negative numbers, converting them into expressions involving \( i \). Learning to work with complex numbers opens the door to solving a wider range of problems, such as those in engineering and physics.

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Most popular questions from this chapter

Use DeMoivre's theorem to find all the indicated roots. Be sure to find all roots. The two square roots of \(4\left(\cos 60^{\circ}+j \sin 60^{\circ}\right)\)

Represent each complex number graphically and give the rectangular form of each. $$220.8\left(\cos 155.13^{\circ}+j \sin 155.13^{\circ}\right)$$

Perform the indicated operations and express results in rectangular and polar forms. $$\left(625 e^{3.46 j}\right)\left(4.40 e^{1.22 j}\right)$$

Answer the given questions. Are \(8 j\) and \(-8 j\) the solutions to the equation \(x^{2}+64=0 ?\)

answer or explain as indicated. Explain how to show that the reciprocal of the imaginary unit is the negative of the imaginary unit.
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