Problem 50 - Algebra 2

Question

Simplify each expression. $$ \frac{x^{\frac{5}{3}}-x^{\frac{1}{3}} z^{\frac{4}{3}}}{x^{\frac{2}{3}}+z^{\frac{2}{3}}} $$

Step-by-Step Solution

Answer

The simplified expression is \(x^{\frac{1}{3}} \cdot \frac{x^{\frac{4}{3}} - z^{\frac{4}{3}}}{x^{\frac{2}{3}} + z^{\frac{2}{3}}}\).

Step 1: Analyze the Expression

First, let's analyze the given expression: \[\frac{x^{\frac{5}{3}}-x^{\frac{1}{3}} z^{\frac{4}{3}}}{x^{\frac{2}{3}}+z^{\frac{2}{3}}}\]We need to simplify the numerator and denominator separately and then the entire fraction.

Step 2: Factor the Numerator

The numerator is given as \[x^{\frac{5}{3}} - x^{\frac{1}{3}} z^{\frac{4}{3}}\].Let's factor out the common term from both expressions, which is \(x^{\frac{1}{3}}\).This gives us:\[x^{\frac{1}{3}}(x^{\frac{4}{3}} - z^{\frac{4}{3}})\].

Step 3: Rewrite the Denominator Using Common Factor

The denominator is \[x^{\frac{2}{3}} + z^{\frac{2}{3}}\].There is no common factor to extract, so we leave it as it is for now.

Step 4: Simplify the Expression

Now, substitute the factored form of the numerator back into the original expression:\[\frac{x^{\frac{1}{3}}(x^{\frac{4}{3}} - z^{\frac{4}{3}})}{x^{\frac{2}{3}} + z^{\frac{2}{3}}}\]

Step 5: Further Simplify if Possible

Since \(x^{\frac{4}{3}} - z^{\frac{4}{3}}\) from the numerator does not further simplify with the denominator \(x^{\frac{2}{3}} + z^{\frac{2}{3}}\), the expression is in its simplest possible form, assuming there are no common terms present. The simplest form of the expression is:\[x^{\frac{1}{3}} \cdot \frac{x^{\frac{4}{3}} - z^{\frac{4}{3}}}{x^{\frac{2}{3}} + z^{\frac{2}{3}}}\].

Key Concepts

Factoring Expressions

Factoring is a fundamental skill in algebra that helps simplify expressions by identifying common elements present in every term of an expression. When you factor an expression, you are essentially "taking out" the common component, simplifying the terms that are left.
In the provided exercise, we started by factoring the numerator of the fraction:

• The original expression had a numerator of \(x^{\frac{5}{3}} - x^{\frac{1}{3}} z^{\frac{4}{3}}\).
• We noticed that both terms shared \(x^{\frac{1}{3}}\) as a factor.
• By factoring \(x^{\frac{1}{3}}\), the expression became \(x^{\frac{1}{3}}(x^{\frac{4}{3}} - z^{\frac{4}{3}})\).

Factoring like this is crucial because it can often expose further opportunities to simplify or reduce the expression, particularly in rational expressions. Look for common factors in both terms, then "extract" them outside a set of parentheses, simplifying calculations and sometimes even simplifying the entire expression down the road.

Fraction Simplification

Simplifying fractions involves making both the numerator and the denominator as small as possible while keeping the quotient the same. In algebra, this often involves factoring both parts of a fraction to cancel out common terms.
Let's look at our expression:

• The factored form of the numerator is \(x^{\frac{1}{3}}(x^{\frac{4}{3}} - z^{\frac{4}{3}})\).
• The denominator is \(x^{\frac{2}{3}} + z^{\frac{2}{3}}\), where there aren't direct factors to simplify further.

At this point, the fraction cannot be simplified further because there are no common factors remaining in the numerator and denominator. Always check if part of the expression could allow for cancellation—you do this by ensuring factorization of both numerator and denominator to see if any terms match.

Exponents and Powers

Exponents help us write repeated multiplication compactly, and understanding how they work is crucial for simplifying many algebraic expressions.
Here are some key rules and applications relevant to the exercise:

• Power of a Power:\((a^m)^n = a^{mn}\). This rule is useful when lifting an already exponentiated variable to another power.
• Product of Powers:\(a^m \cdot a^n = a^{m+n}\). This rule helps combine like bases with exponents.
• Division of Powers:\(\frac{a^m}{a^n} = a^{m-n}\) for simplifying when dividing like bases.

In the exercise, we see exponents presented in fractional form. With fractional exponents, the numerators provide the power to which the base number is raised and the denominators indicate the root of the base (e.g., \(x^{\frac{5}{3}}\) means \(x\) is raised to the fifth power, then the cube root is taken).
This intermediary form of exponent representation can simplify calculations and manipulations, especially when dealing with radicals in algebraic expressions.