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Chapter 12: Problem 27

Simplify each of the given expressions. $$j^{2}-j^{6}$$

Short Answer

Expert verified
The expression simplifies to \( j^2(1 - j^4) \).

Step by step solution

01

Factor out the common term

Identify the common factor in the expression. In this case, both terms share the factor \( j^2 \). By factoring \( j^2 \) out of the expression, rewrite the expression as: \[ j^2(1 - j^4) \].
02

Simplify the expression inside the parenthesis

The expression \( 1 - j^4 \) does not simplify further using common algebraic identities, so we keep it as is. Thus, the simplified expression remains: \[ j^2(1 - j^4) \].

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Key Concepts

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

Factoring
Factoring is a fundamental concept in algebra that involves finding common terms in an expression and rewriting it as a product of simpler expressions. When you factor an expression, you are essentially breaking it down into parts that are easier to work with.
For example, in the exercise, we start with the expression \( j^{2} - j^{6} \). Both terms have a common factor of \( j^{2} \).
  • We "factor out" \( j^{2} \), meaning we divide each term by \( j^{2} \) and write the expression as a product: \( j^{2}(1 - j^{4}) \).
  • This process simplifies the expression and makes it easier to handle.

Factoring is a crucial skill for simplifying complex expressions and solving equations efficiently. It's like uncovering the layers of an onion, getting to the simpler core within.
Exponents
Exponents tell us how many times to multiply a number by itself. In the expression \( j^{2} - j^{6} \), the numbers 2 and 6 are the exponents.
  • When dealing with exponents, a basic rule is that when you multiply like bases, you add their exponents: \( a^{m} \times a^{n} = a^{m+n} \).
  • Similarly, when dividing like bases, you subtract the exponents: \( \frac{a^{m}}{a^{n}} = a^{m-n} \).
In our factoring step, by recognizing \( j^{2} \) as a common term, we effectively used these rules.
Understanding exponents is vital because they appear often in algebraic simplification, allowing us to handle and streamline expressions more effectively.
Algebraic Expressions
Algebraic expressions are made up of variables, numbers, and operations like addition and subtraction. An important aspect of working with algebraic expressions is recognizing how to manipulate these parts to simplify or solve them.
  • In the expression \( j^{2} - j^{6} \), the variable is \( j \), and the operations include subtraction and exponentiation.
  • Simplifying algebraic expressions involves combining like terms, factoring, and reducing the expression to its simplest form.
Knowing how to break down and simplify algebraic expressions is crucial for solving equations and understanding the relationships between variables.
Practice analyzing and rewriting these expressions to gain familiarity and confidence in using algebraic techniques.

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Most popular questions from this chapter

Express the given complex numbers in polar and rectangular forms. $$0.800 e^{3.00 j}$$

$$\text {solve the given problems.}$$ Write \(j^{-2}+j^{-3}\) in rectangular form.

Use DeMoivre's theorem to find all the indicated roots. Be sure to find all roots. The square roots of \(1+j\)

answer or explain as indicated. What type of number is the result of (a) adding a complex number to its conjugate and (b) subtracting a complex number from its conjugate?

Represent each complex number graphically and give the rectangular form of each. $$3.00\left(\cos 232.0^{\circ}+j \sin 232.0^{\circ}\right)$$
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