Problem 11 - Contemporary Precalculus

Question

Compute and simplify. $$x^{1 / 2}\left(x^{2 / 3}-x^{4 / 3}\right)$$

Step-by-Step Solution

Answer

Question: Simplify the expression: \(x^{1 / 2}(x^{2 / 3}-x^{4 / 3})\). Answer: The simplified expression is: \(x^{\frac{7}{6}} - x^{\frac{11}{6}}\).

Step 1: Apply the exponent rule for multiplication

To simplify the given expression, we will first apply the exponent rule for multiplication, i.e., when multiplying two powers with the same base, we add the exponents. In this case, the base is x. $$x^{1 / 2}\left(x^{2 / 3}-x^{4 / 3}\right)$$ Let's multiply each term inside the parentheses by \(x^{1/2}\).

Step 2: Multiply the terms with a common base

We multiply \(x^{1 / 2}\) with each term inside the parentheses and apply the exponent rule for multiplication, i.e., adding the exponents. $$x^{1 / 2} \cdot x^{2 / 3} - x^{1 / 2} \cdot x^{4 / 3}$$ This results in: $$x^{\frac{1}{2}+\frac{2}{3}} - x^{\frac{1}{2}+\frac{4}{3}}$$ Now, compute the new exponents.

Step 3: Compute the new exponents

Now, calculate the sum of the exponents: $$\frac{1}{2}+\frac{2}{3} = \frac{3+4}{6} = \frac{7}{6}$$ $$\frac{1}{2}+\frac{4}{3} = \frac{3+8}{6} = \frac{11}{6}$$ Therefore, the simplified expression becomes: $$x^{\frac{7}{6}} - x^{\frac{11}{6}}$$

Step 4: Write the final expression

Now that we have simplified the given expression, we can rewrite the final expression: $$x^{1 / 2}\left(x^{2 / 3}-x^{4 / 3}\right) = x^{\frac{7}{6}} - x^{\frac{11}{6}}$$ So, the simplified expression is: $$x^{\frac{7}{6}} - x^{\frac{11}{6}}$$

Key Concepts

Exponent Rules

Exponents play a crucial role in algebra, providing a shorthand way to express repeated multiplication. Understanding exponent rules is essential for simplifying algebraic expressions. One foundational rule is the multiplication of powers with the same base: when you multiply two exponential terms with the same base, you keep the base and add the exponents together. It looks like this: \( x^a \cdot x^b = x^{a+b} \). This rule is particularly important when simplifying expressions with multiple exponential terms.

For instance, if given \( x^{1/2} \cdot x^{2/3} \), we can quickly use this rule to find that \( x^{1/2 + 2/3} \), which simplifies further into \( x^{7/6}\). The same rule applies to expressions with several terms, such as \( x^{1/2} \cdot x^{2/3} - x^{1/2} \cdot x^{4/3} \) leading to \( x^{7/6} - x^{11/6} \) after simplification.

In practice, to apply this rule successfully, familiarity with adding fractions is also important, as we often deal with fractional exponents in algebra.

Multiplication of Powers

The multiplication of powers is not only limited to exponents with a common base, it also includes situations where variables and numbers are combined. This particular aspect challenges students to recognize that the same exponent rules apply when working with algebraic expressions that may seem more complicated.

Using the distributive property, one might encounter expressions like \( 3x^2 \cdot 4x^3 \), in which both coefficients and powers of x can be multiplied together. By applying exponent rules, \( 3 \cdot 4 \) becomes 12, and \( x^2 \cdot x^3 \) becomes \( x^{2+3} \) or \( x^5 \) using the rule for multiplying powers with the same base. The resulting expression is \( 12x^5 \).

Remember that these principles do not just apply to numerical bases; variables are subject to the same laws of exponents. This fundamental part of algebraic simplification helps to propel students from understanding basic arithmetic to mastering more complex algebraic manipulation.

Algebraic Simplification

Algebraic simplification refers to the process of rewriting an expression in its simplest, most concise form. It involves combining like terms, reducing fractions, and applying the exponent rules we've already discussed.

A key step in algebraic simplification is recognizing like terms—which are terms in an algebraic expression that have the same variables with the same exponents. These can be combined by adding or subtracting their coefficients. For example, in the expression \( 3x^2 + 5x^2 \), both terms are 'like' because they have the same variable to the same power; hence, they can be combined to \( 8x^2 \).

When simplifying expressions, it's also important to apply the order of operations correctly, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). By following these rules and understanding the structure of algebraic expressions, students can streamline expressions to reveal a clearer path to the solution.