Problem 50 - Precalculus Mathematics for Calculus

Question

Perform the indicated operations and simplify. $$x^{3 / 2}(\sqrt{x}-1 / \sqrt{x})$$

Step-by-Step Solution

Answer

The simplified expression is \(x^2 - x\).

Step 1: Distribute the expression

First, distribute the outer term \(x^{3/2}\) to both terms inside the parenthesis. This gives us two separate terms: \(x^{3/2} \cdot \sqrt{x}\) and \(x^{3/2} \cdot \frac{1}{\sqrt{x}}\).

Step 2: Simplify each term

For the first term \(x^{3/2} \cdot \sqrt{x}\), recall that \(\sqrt{x}\) is \(x^{1/2}\). So, \(x^{3/2} \cdot x^{1/2} = x^{(3/2 + 1/2)} = x^2\). For the second term \(x^{3/2} \cdot \frac{1}{\sqrt{x}}\), \(\frac{1}{\sqrt{x}}\) is \(x^{-1/2}\). So, \(x^{3/2} \cdot x^{-1/2} = x^{(3/2 - 1/2)} = x^{1}\).

Step 3: Combine and simplify the expression

Combine the simplified terms from the previous step. You get \(x^2 - x^1 = x^2 - x\). This is the simplified form of the original expression.

Key Concepts

Exponent Rules

Understanding exponent rules is essential for dealing with algebraic expressions. Exponents are a shorthand way of expressing repeated multiplication. For example, \( x^3 \) means \( x \) multiplied by itself three times. There are several key rules of exponents that can help simplify expressions:

• Product of Powers Rule: When multiplying two expressions with the same base, you add the exponents. For instance, \( x^a \cdot x^b = x^{a+b} \).
• Quotient of Powers Rule: When dividing two expressions with the same base, you subtract the exponents. For example, \( x^a \div x^b = x^{a-b} \).
• Power of a Power Rule: When taking an exponent to another exponent, you multiply the exponents. For example, \( (x^a)^b = x^{a \cdot b} \).

Recognizing these rules can make solving even complex expressions much easier. In our original problem, we used the product of powers rule to simplify \( x^{3/2} \cdot x^{1/2} \) by adding the exponents, resulting in \( x^2 \). Understanding these rules allows us to manipulate and simplify expressions accurately.

Simplifying Expressions

Simplifying algebraic expressions involves combining like terms and putting the expression in its most reduced form. The goal is to make the expression as simple as possible for easy understanding and use.

• Combining Like Terms: Like terms are terms that have the same variable raised to the same power. You simplify expressions by adding or subtracting these terms.
• Distributive Property: This property allows you to multiply a term across terms inside parentheses. For example, a(b + c) = ab + ac.

In the given exercise, we first distributed the term \( x^{3/2} \) to the terms inside the parentheses, resulting in two new terms, \( x^{3/2} \cdot x^{1/2} \) and \( x^{3/2} \cdot x^{-1/2} \). We then used exponent rules to simplify these terms into \( x^2 \) and \( x^1 \.\) The final step in simplifying was to combine these into the simplified expression \( x^2 - x \). Taking your time with each term to simplify fully helps ensure accuracy.

Operations with Binomials

Operations with binomials, which are algebraic expressions containing two terms, often involve using key algebraic properties to simplify or expand the expression. The process may include:

• Distribution: Applying the distributive property to multiply terms across the binomial, such as multiplying a single term by each term in a binomial.
• Addition and Subtraction: Combining like terms after distribution or multiplication.

In the original problem, the expression \( x^{3/2}(\sqrt{x} - \frac{1}{\sqrt{x}}) \) is simplified by distributing \( x^{3/2} \) to each term within the binomial. Mastering the distribution across binomials is useful in algebra as it appears frequently in expansion of polynomials and solving equations. By practicing distributing carefully, you start handling more complex expressions accurately.