/*! 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 82 Find each product. $$ [(4 k+... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 82

Find each product. $$ [(4 k+h)-4]^{2} $$

Short Answer

Expert verified
\[ [(4k + h) - 4]^2 = 16k^2 + 8kh + h^2 - 8k - 8h + 16 \]

Step by step solution

01

Recognize the Binomial Square

The expression needs to be evaluated as a binomial square. Recognize that \( (a - b)^2 \) can be expanded using the formula \( a^2 - 2ab + b^2 \). For the given expression \( [(4k + h) - 4]^2 \), identify \( a = 4k + h \) and \( b = 4 \).
02

Expand the Binomial

Using the binomial expansion formula \ ((a - b)^2 = a^2 - 2ab + b^2) \, expand the given expression: \[ (4k + h - 4)^2 = (4k + h)^2 - 2(4k + h)(4) + 4^2 \]
03

Compute the Individual Terms

Compute each term of the expansion separately: \( (4k + h)^2 \), \( -2(4k + h)(4) \), and \( 4^2 \). 1. \( (4k + h)^2 = 16k^2 + 8kh + h^2 \) 2. \( -2(4k + h)(4) = -8k - 8h \) 3. \( 4^2 = 16 \)
04

Combine All Terms

Combine all the terms obtained from the previous step: \[ (4k + h - 4)^2 = 16k^2 + 8kh + h^2 - 8k - 8h + 16 \]

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.

binomial square
The expression given in the problem \([ (4k + h) - 4 ]^2\) is known as a binomial square. In simple terms, a binomial square involves squaring a binomial expression. A binomial expression is an algebraic expression that contains two terms.
When we square a binomial, we use a specific formula to simplify it. The formula for squaring a binomial is \((a - b)^2 = a^2 - 2ab + b^2\).
Understanding how to recognize and work with binomial squares is essential in algebra, as it allows us to expand and simplify expressions quickly and accurately. In this case, recognizing the structure of the binomial square helps us identify the parts of the expression that need to be expanded.
algebraic expression
An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. In the given exercise, \[ (4k + h - 4)^2 \], \[(4k + h - 4) \] is an algebraic expression. It includes:
  • Numbers (4, in this case)
  • Variables (like k and h)
  • Operations (like addition and subtraction)
Understanding the structure of algebraic expressions is crucial because it allows us to apply different mathematical rules and operations correctly. For example, once we recognize the form of the expression, we can apply the binomial expansion formula to simplify it.
expansion formula
The binomial expansion formula is used to expand expressions that are raised to a power. For binomials like \[(a - b)^2\], the formula is \(a^2 - 2ab + b^2\).
Here are the steps to use the formula:
  • Identify the terms in the binomial. For the given expression \[(4k + h - 4)^2\], we have \(a = 4k + h\) and \(b = 4\).
  • Apply the formula. Substitute \(a\) and \(b\) into the formula: \[(4k + h)^2 - 2(4k + h)(4) + 4^2\]
  • Simplify the individual terms. Calculate each part separately:
    1. \((4k + h)^2 = 16k^2 + 8kh + h^2\)
    2. \(-2(4k + h)(4) = -8k - 8h\)
    3. \(4^2 = 16\)
  • Combine the terms. Add all the simplified terms together: \[16k^2 + 8kh + h^2 - 8k - 8h + 16\]
Using this formula makes it straightforward to break down and expand any binomial square.

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

Find each product. $$ (2 a+b)\left(3 a^{2}+2 a b+b^{2}\right) $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(\frac{8 x^{-6} y^{3}}{x^{4} y^{-4}}\right)^{-2} $$

Find each product. $$ [(2 h-k)+j][(2 h-k)-j] $$

Find each product. $$ (5 r-s)^{3} $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(x^{-3}\right)^{6} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.