/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 106 Perform the indicated operations... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 106

Perform the indicated operations. a. \((c d)^{2}\) b. \((c-d)^{2}\)

Short Answer

Expert verified
a. \(c^2 d^2\) b. \(c^2 - 2cd + d^2\)

Step by step solution

01

Understanding the Problem

We need to perform operations on two expressions. Part (a) involves squaring a product of terms \( (c d)^2 \), and part (b) involves squaring a binomial \( (c - d)^2 \).
02

Squaring the Product for Part (a)

To square the product \((c d)\), apply the rule \((xy)^2 = x^2 y^2\). Here, \(x = c\) and \(y = d\), so:\[(c d)^2 = c^2 d^2\].
03

Squaring the Binomial for Part (b)

To square the binomial \((c - d)\), use the binomial square expansion \((x - y)^2 = x^2 - 2xy + y^2\). Here, \(x = c\) and \(y = d\), so:\[(c - d)^2 = c^2 - 2cd + d^2\].
04

Writing the Final Answers

For part (a), the result is \(c^2 d^2\). For part (b), the result is \(c^2 - 2cd + d^2\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Squaring a Product
When we talk about squaring a product, we're dealing with the expression \((c d)^2\). This means you take the product of two numbers or variables and then square it. Simply put, you use the rule: \((xy)^2 = x^2 y^2\).

This rule teaches us that to square a product, you square each factor individually and then multiply them back together. For our example,
  • Set \(x = c\) and \(y = d\).
  • Apply the rule to get \((c d)^2 = c^2 d^2\).
The result shows that each term in the product, here \(c\) and \(d\), is squared separately before being multiplied together. This fundamental property helps simplify expressions and solve algebraic equations efficiently.
Squaring a Binomial
Squaring a binomial involves working with an expression like \((c - d)^2\). A binomial is simply a two-term expression connected by either addition or subtraction. To square a binomial, you use the formula:\((x - y)^2 = x^2 - 2xy + y^2\).

Using this formula simplifies the process:
  • Identify \(x = c\) and \(y = d\).
  • Apply the binomial expansion to attain \((c - d)^2 = c^2 - 2cd + d^2\).
This formula reveals that squaring a binomial doesn't just square each term; the middle term \(-2xy\) results from the cross-multiplication of \(x\) and \(y\). Recognizing this pattern is crucial, as it appears frequently in more complex algebraic equations.
Binomial Expansion
Binomial expansion covers methods for simplifying powers of a binomial. In essence, when you square a binomial like \((x - y)^2\), it means multiplying the binomial by itself. However, instead of manually multiplying and combining terms, you use the formula. This kind of expansion is vital for understanding how larger powers of binomials unfold.

When doing a binomial expansion, think about how each component behaves:
  • The square of the first term \(x^2\).
  • The product of the terms, doubled: \(-2xy\).
  • The square of the second term \(y^2\).
Expansion helps simplify work, keeping calculations clean and problem-solving manageable. This understanding not only aids in algebra but lays the foundation for polynomial expansions, crucial in calculus and advanced algebra.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

a. Fill in the blanks: \((x y)^{2}\) is the _____ of a product and \((x+y)^{2}\) is the _____ of a sum. b. Explain why \((x y)^{2} \neq(x+y)^{2}\).

Perform the operations. $$ \left(6-5 p^{2}\right)^{2} $$

Communications. Telephone poles were installed every \((2 x-3)\) feet along a stretch of railroad track \(\left(8 x^{3}-6 x^{2}+5 x-21\right)\) feet long. What expression represents the number of poles that were used?

Perform each division. $$ \frac{a^{3}+a}{a+3} $$

Perform each division. $$ \frac{8 a^{2}+2 a-3}{2 a-1} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.