Problem 93 Perform the operations. $$ (... [FREE SOLUTION]

Chapter 5: Problem 93

Perform the operations. $$ (f-8)^{2} $$

Short Answer

Expert verified

The expanded form is $f^2 - 16f + 64$.

Step by step solution

Understand the Problem

We need to expand the expression $(f-8)^2$. This expression is a binomial squared, meaning it follows the structure $(a-b)^2 = a^2 - 2ab + b^2$.

Identify Parts of the Binomial

In the expression $(f-8)^2$, identify $a = f$ and $b = 8$. These values will be used in the binomial expansion formula.

Apply the Binomial Formula

Using the formula $(a-b)^2 = a^2 - 2ab + b^2$, substitute $a = f$ and $b = 8$. This gives $f^2 - 2(f)(8) + 8^2$.

Perform Each Calculation

Calculate each term: - $f^2$ stays as $f^2$.- $-2(f)(8) = -16f$.- $8^2 = 64$.

Write the Final Expanded Form

Combine the terms to get the expanded expression: $f^2 - 16f + 64$.

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Key Concepts

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

Understanding Algebra in Binomial Expansion

Algebra plays a crucial role in simplifying expressions like $(f-8)^2$. At its core, algebra is about finding and applying rules to manipulate mathematical symbols. This simplification process makes complex problems more approachable. Here, the task is to expand a binomial expression, which involves several algebraic concepts.

When working with algebra:

Recognize patterns in expressions, such as binomials where terms are added or subtracted.
Apply known formulas to expand, factor, or simplify expressions efficiently.

Recognizing the expression $(f-8)^2$ as a binomial square is an essential algebraic skill. By breaking it down, $(a-b)^2 = a^2 - 2ab + b^2$, we can systematically tackle the expansion by recognizing the roles of $a$ and $b$. Algebra helps us turn a seemingly complex task into a series of simpler steps.

The Binomial Theorem and Its Application

The binomial theorem is a cornerstone in expanding expressions involving powers of binomials. The theorem provides a standardized way of finding expanded forms, saving time and effort over more laborious manual multiplication.

Specifically, the binomial theorem gives us a blueprint for expanding squares of two-term expressions of the form $(a-b)^2$. It states:

First term squared: $a^2$
Twice the product of both terms: $-2ab$
Second term squared: $b^2$

By identifying $a = f$ and $b = 8$ in $(f-8)^2$, we can easily plug these into the formula to get the expanded expression $f^2 - 16f + 64$. The binomial theorem simplifies the process to ensure accuracy and efficiency in solving such problems.

Performing Mathematical Operations in Binomial Expansion

When expanding a binomial expression, it's crucial to perform each mathematical operation accurately. The expansion $(f-8)^2$ is executed step by step:

First, identify the individual components:

$a = f$
$b = 8$

Next, execute the mathematical operations dictated by the binomial squared formula:

Calculate $f^2$.
Find $-2(f)(8)$, which results in $-16f$.
Compute $8^2$, resulting in $64$.

Finally, combine these computed terms to achieve the expanded expression: $f^2 - 16f + 64$.

This meticulous step-by-step usage of mathematical operations ensures that the expanded form of the binomial is accurate and correctly accounted for each part of the original expression.

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91影视

Perform the operations. $$ (f-8)^{2} $$

Short Answer

Step by step solution

Understand the Problem

Identify Parts of the Binomial

Apply the Binomial Formula

Perform Each Calculation

Write the Final Expanded Form

Key Concepts

Understanding Algebra in Binomial Expansion

The Binomial Theorem and Its Application

Performing Mathematical Operations in Binomial Expansion

One App. One Place for Learning.

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