Problem 85 - Algebra and Trigonometry

Question

Perform the indicated operations, and simplify. \(\left(1+x^{2 / 3}\right)\left(1-x^{2 / 3}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(1 - x^{4/3}\).

Step 1: Recognize the Expression Type

The expression \( (1+x^{2/3})(1-x^{2/3}) \) is of the form \((a+b)(a-b)\), which is a difference of squares identity. Recall that \((a+b)(a-b) = a^2 - b^2\).

Step 2: Identify Parts of the Expression

Identify \( a = 1 \) and \( b = x^{2/3} \), meaning the expression becomes \((1)^2 - (x^{2/3})^2\).

Step 3: Apply the Difference of Squares

Using the formula \((a+b)(a-b) = a^2 - b^2\), substitute the values identified in Step 2 into the formula. This gives \(1^2 - (x^{2/3})^2 = 1 - x^{4/3}\).

Step 4: Simplify the Expression

Since \(1^2 = 1\), writing the result explicitly, we have the expression simplified as \(1 - x^{4/3}\).

Key Concepts

Difference of Squares

The difference of squares is a helpful identity in mathematics that simplifies specific polynomial expressions. This identity is applied when you have two terms that are added together in one bracket and subtracted in another. For example, something like \((a+b)(a-b)\). Here, the result is always the square of \(a\) minus the square of \(b\). Simplified, it reads:

• \(a^2 - b^2\)

In the original problem, the expression \((1+x^{2/3})(1-x^{2/3})\) follows this form. By recognizing it as the difference of squares, you can apply the formula directly:

• Identify \(a = 1\) and \(b = x^{2/3}\).
• Apply the identity: \((1)^2 - (x^{2/3})^2\).
• Simplifies to \(1 - x^{4/3}\).

This method is a handy tool to simplify expressions and solve algebraic problems quickly.

Rational Exponents

Rational exponents can be tricky at first, but once you break them down, they start to make sense. A rational exponent is just another way to represent roots and powers using fractions. The numerator indicates the power, while the denominator represents the root. For instance, \(x^{2/3}\) tells you:

• Square \(x\) (because of the 2 in the numerator).
• Then take the cube root (because of the 3 in the denominator).

This operation can be simplified further. Sometimes, you might see something like \((x^{2/3})^2\), and realize it is simplest using:
\((x^{2/3})^2 = x^{4/3}\),where you multiply the exponents. In algebra, handling rational exponents adeptly is vital. It allows for smoother operations and easier simplifications.

Algebraic Simplification

Algebraic simplification is all about making expressions cleaner and more manageable. The goal is to represent an expression in its simplest form by combining like terms, reducing expressions, and using identities. In the expression \(1 - x^{4/3}\), notice no further simplification can be done as there are no like terms to combine.

• Recognize patterns and identities like the difference of squares.
• Apply rational exponents rules to manipulate terms.
• Ensure each term is in its simplest form.

By following these methods, you can transform a potentially complex expression into something more intuitive and easy to understand. Practice is key to mastering this skill, just like simplifying our original problem!