Problem 83 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand and simplify the expression. \(\left(x^{2 / 3}-y^{1 / 3}\right)^{3}\)

Step-by-Step Solution

Answer

The expanded and simplified form of the expression \( \left(x^{2 / 3}-y^{1 / 3}\right)^{3} \) is \( x^2 -3*x^{4/3} y^{1/3} + 3x^{2/3} y^{2/3} - y \)

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that, for any positive integer \(n\), \[ (a+b)^n = \sum_{k=0}^{n} {n\choose k} a^{n-k} * b^{k} \] This where \( {n\choose k} \) represents a binomial coefficient, which is calculated as \( \frac{n!}{k!(n-k)!} \). When performing a cube expansion (where \( n = 3 \)), the expanded form would be: \[ (a+b)^3 = a^3 + 3a^2*b + 3a*b^2 + b^3 \]

Step 2: Apply Binomial Theorem according to the exercise

In this exercise, given \( a = x^{2/3} \) and \( b = -y^{1/3} \), we apply the binomial theorem to get the equivalent of \( \left(x^{2 / 3}-y^{1 / 3}\right)^{3} \) \[ \left(x^{2 / 3}-y^{1 / 3}\right)^{3} = (x^{2/3})^3 + 3(x^{2/3})^2 * (-y^{1/3}) + 3(x^{2/3}) * (-y^{1/3})^2 +(-y^{1/3}) ^3 \]

Step 3: Simplify the Result

Now we simply calculate those expressions to get the simplified form of the binomial expansion: \[ \left(x^{2 / 3}-y^{1 / 3}\right)^{3} = x^2 -3*x^{4/3} y^{1/3} + 3x^{2/3} y^{2/3} - y \]

Key Concepts

Understanding Binomial Coefficients

In the realm of algebra, binomial coefficients play a vital role, especially when using the Binomial Theorem. A binomial coefficient, denoted as \( \binom{n}{k} \), helps us determine the coefficients in the expansion of a binomial raised to a power. These coefficients are defined as \( \frac{n!}{k!(n-k)!} \), where \(!\) denotes factorial. For example, in \((a+b)^3\), the binomial coefficients are 1, 3, 3, and 1, as seen in the expanded form:

• \(a^3\)
• \(3a^2b\)
• \(3ab^2\)
• \(b^3\)

Understanding how to calculate and apply these coefficients is crucial for expanding and simplifying expressions using the Binomial Theorem.

Cube Expansion with the Binomial Theorem

Expanding binomials to the third power involves using the Binomial Theorem, a fundamental concept in algebra. The expression \((a+b)^3\) can be expanded as \(a^3 + 3a^2b + 3ab^2 + b^3\). This method allows us to manage expressions involving powers, helping simplify complex algebraic expressions efficiently.
Applying this to \((x^{2/3}-y^{1/3})^3\), we substitute \(a = x^{2/3}\) and \(b = -y^{1/3}\):

• First term: \((x^{2/3})^3 = x^2\)
• Second term: \(3(x^{2/3})^2(-y^{1/3}) = -3x^{4/3}y^{1/3}\)
• Third term: \(3(x^{2/3})(-y^{1/3})^2 = 3x^{2/3}y^{2/3}\)
• Fourth term: \((-y^{1/3})^3 = -y\)

This simplifies to \(x^2 - 3x^{4/3}y^{1/3} + 3x^{2/3}y^{2/3} - y\). Understanding this expansion makes solving related problems more intuitive.

Simplifying Expressions

Simplifying algebraic expressions is a core skill that aids in managing complex mathematical problems. When simplifying, the aim is to condense the expression while maintaining equivalence. In the context of binomial expansion, it involves combining like terms and simplifying exponents.In our example, after expanding \((x^{2/3}-y^{1/3})^3\), we're left with terms such as \(x^2\), \(-3x^{4/3}y^{1/3}\), \(3x^{2/3}y^{2/3}\), and \(-y\). Here are some key points to simplify:

• Identify and combine like terms, which might involve common variables or similar powers.
• Reduce complex fractions and expressions when possible.
• Ensure the expression is concise, making it easier to interpret and apply in further calculations.

Successfully simplifying expressions allows for better clarity and understanding of the underlying mathematical relationships.