Problem 51 Perform the indicated operations... [FREE SOLUTION]

Chapter 0: Problem 51

Perform the indicated operations and simplify. $$ (w+4)^{3} $$

Short Answer

Expert verified

The simplified form is \ w^3 + 12w^2 + 48w + 64 \.

Step by step solution

- Expand the Cubic Expression

To expand $ (w+4)^3 $, use the binomial theorem, which states \[ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]. For this exercise, \ a = w \ and \ b = 4 \.

- Apply the Binomial Theorem

Substitute \ a = w \ and \ b = 4 \ into the binomial theorem: \[ (w + 4)^3 = w^3 + 3(w^2)(4) + 3(w)(4^2) + 4^3 \].

- Calculate Each Term

Evaluate each term in the expression: \[ w^3 + 3(16w^2) + 3(4^2w) + 4^3 \] simplifies to \[ w^3 + 12w^2 + 48w + 64 \].

- Combine the Terms

Write the final expanded and simplified form: \[ w^3 + 12w^2 + 48w + 64 \].

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Key Concepts

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

Cubic Expression Expansion

To fully understand this exercise, we need to grasp cubic expression expansion.
A cubic expression refers to any expression raised to the power of three. In our case, it is $ (w+4)^3$. Expanding such expressions involves using mathematical formulas designed for higher powers.
The binomial theorem is a powerful tool for expanding any binomial raised to a power. Here, particularly, we use it for cubic (^3) expansions.
By substituting the terms into the formula $ (a+b)^3$ and calculating systematically, we transform complicated expressions into more workable ones.
As a result, \[ (w+4)^3 = w^3 + 12w^2 + 48w + 64 \].

Polynomial Simplification

Simplifying polynomials is another critical aspect of this problem,
once expansions are completed.
When we expanded the cubic expression, we initially got multiple terms:
For instance, $ w^3, 12w^2, 48w, \, \text{and}\, 64 $. Simplification means combining like terms and ensuring that the polynomial is written in its simplest form.
They were already simple, distinct terms here, so the next step was to write them clearly:
Thus, \[ w^3 + 12w^2 + 48w + 64 \] is in its simplest form when there are no further operations left.

Binomial Expansion

Here is a step-by-step explanation of binomial expansion:
The binomial theorem is crucial for expanding expressions of form $ (a+b)^n$, where $ n $ is an integer.
For a cubic expression $ (a+b)^3$, it expands as \[ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \].
Let's plug in $ w $ for $ a $ and $ 4 $ for $ b $:
\[ (w+4)^3 \ \Rightarrow w^3 + 3w^2(4) + 3w(4^2) + 4^3 \].
The steps following involve calculating individual terms:

$ w^3 = w^3 $
$ 3w^2(4) = 12w^2 $
$ 3w(16) = 48w $
$ 4^3 = 64 $.

Finally, you combine to get:
\in wrright3 + 12wrright2 + 48wr plus&64 .

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91影视

Perform the indicated operations and simplify. $$ (w+4)^{3} $$

Short Answer

Step by step solution

- Expand the Cubic Expression

- Apply the Binomial Theorem

- Calculate Each Term

- Combine the Terms

Key Concepts

Cubic Expression Expansion

Polynomial Simplification

Binomial Expansion

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