Problem 64 - Fundamentals of Mathematics

Question

Expand \(84^{3}\). Do not find the value.

Step-by-Step Solution

Answer

Expanded form: \(80^3 + 3(80^2)(4) + 3(80)(4^2) + 4^3\).

Step 1: Understand the Cube Expansion Formula

To expand \(a^3\), we use the formula \(a^3 = (a)^3 = a \times a \times a\). For a binomial \((x + y)^3\), the expanded form is \(x^3 + 3x^2y + 3xy^2 + y^3\). But here, we use \(a^3 \) directly, and only need multiplication to expand into steps without calculating the final product.

Step 2: Rewrite Using Base Numbers

The number 84 can be rewritten as \((80 + 4)\) to facilitate expansion using the binomial theorem. Here, \(x = 80\) and \(y = 4\).

Step 3: Apply Binomial Expansion

Using the binomial theorem expansion for \((x + y)^3\),\( (80 + 4)^3 = 80^3 + 3(80^2)(4) + 3(80)(4^2) + 4^3 \). This is the expanded form of \(84^3\).

Step 4: Write Each Term

Identify each term from the expanded form of \((80+4)^3\):1. \(80^3\) is just the cube of the first number.2. \(3(80^2)(4)\) is three times the square of the first term times the second term.3. \(3(80)(4^2)\) is three times the first term times the square of the second term.4. \(4^3\) is the cube of the second term.

Key Concepts

Cube Expansion

Expanding a binomial cube involves taking a number to the third power, like finding the volume of a cube with side length equal to the number. In this exercise, we are looking at the number 84, which means we're figuring out the expression for how it multiplies with itself three times. The formula for the cube expansion is usually presented in the way of

• Taking a single term and multiplying it by itself three times, like this: \[a^3 = a \times a \times a\]
• Or, for a binomial like \((x + y)^3\), it expands into \(x^3 + 3x^2y + 3xy^2 + y^3\)

For the specific case of 84, we do not immediately find the final total, but rather we make use of the binomial expansion to break this down into simpler parts. By doing so, it becomes easier to handle larger calculations without overwhelming arithmetic. By breaking down 84 into \(80 + 4\), we leverage upon known formulas, enabling easier mental or paper calculations later.

Binomial Expansion

The concept of binomial expansion is used to expand expressions that are raised to a power. In this exercise, we use it to expand \((80 + 4)^3\) because 84 is rewritten as \(80 + 4\) to facilitate easier calculation. First, understand the Binomial Theorem. When you raise a binomial to power three, you obtain four terms using:

• \(x^3\): The cube of the first term.
• \(3x^2y\): Three times the square of the first term times the second term.
• \(3xy^2\): Three times the first term times the square of the second term.
• \(y^3\): The cube of the second term.

For \((80 + 4)^3\), this leads to:- \(80^3\)- \(3(80^2)(4)\)- \(3(80)(4^2)\)- \(4^3\)By organizing and calculating these components separately, you keep track of numbers more efficiently while still following the same reliable algebraic formulas. Thus, the binomial expansion simplifies what initially looks like a complex multiplication problem.

Mathematics Formulae

Mathematics is filled with powerful formulae that simplify complex calculations. In this problem, we employ such formulae to structure the expansion. Formulas help streamline processes and improve calculation accuracy, an essential aspect in mathematics where consistency and precision are key. Some of the formulas to note include:

• The Cube Formula: Used to simplify and represent multiplication of a number by itself thrice, as seen in the exercise with \(a^3 = a \times a \times a\)
• The Binomial Formula: Allows the expansion of any binomial raised to a power, breaking it into sum of terms, e.g., \((x + y)^3\)

Applying these formulas does not just make calculations manageable; it transforms how problems are approached. You naturally break down a daunting multiplication into smaller, more handleable steps. Using these formulas, students gain clearer insights into algebraic processes and can solve similar problems with greater confidence.