Problem 76 - Algebra and Trigonometry

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: Expand the Expression

First, use the distributive property to multiply the expression inside the parentheses by \(x^{3/2}\):\[x^{3/2} \cdot \sqrt{x} - x^{3/2} \cdot \frac{1}{\sqrt{x}}\].

Step 2: Simplify the First Term

Simplify \(x^{3/2} \cdot \sqrt{x}\):Recall that \(\sqrt{x} = x^{1/2}\). Therefore, the term becomes:\[x^{3/2} \cdot x^{1/2} = x^{(3/2 + 1/2)} = x^2\].

Step 3: Simplify the Second Term

Simplify \(x^{3/2} \cdot \frac{1}{\sqrt{x}}\):This becomes:\[x^{3/2} \cdot x^{-1/2} = x^{(3/2 - 1/2)} = x\].

Step 4: Combine the Simplified Terms

Write the expression by combining the simplified terms from Steps 2 and 3:\[x^2 - x\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps break down expressions into simpler parts. It's used when you want to multiply a single term by each term within parentheses. For example, in the expression \(x^{3/2}( \sqrt{x} - 1/\sqrt{x})\), the distributive property allows us to expand it into:\(x^{3/2} \cdot \sqrt{x} - x^{3/2} \cdot \frac{1}{\sqrt{x}}\). This step makes it easier to tackle complex algebraic operations by handling each part separately. Remember that the distributive property states: \(a(b + c) = ab + ac\). This property is essential for simplifying expressions and will be particularly helpful in advanced algebra problems.

Exponent Rules

Exponent rules govern how we handle powers of numbers, helping to simplify expressions involving exponents. One key rule used in our original problem is that when you multiply like bases, you add the exponents: \(x^a \cdot x^b = x^{a+b}\).
For example, in the first term \(x^{3/2} \cdot \sqrt{x}\), we interpret \(\sqrt{x}\) as \(x^{1/2}\), then apply the rule to get: \(x^{3/2+1/2} = x^2\). Exponent rules are crucial in algebra, allowing us to transform complicated problems into simple, solvable expressions. Another common rule is \(x^a/x^b = x^{a-b}\), which is used in the second term of our process, \(x^{3/2} \cdot x^{-1/2}\), simplifying to \(x\). Understanding these rules will make algebraic manipulation much more intuitive.

Radicals

Radicals represent roots of numbers, such as square roots. They can sometimes be more challenging to work with, but certain rules make them manageable. In algebra, \(\sqrt{x}\) is often rewritten as \(x^{1/2}\) for simplicity, especially when combined with exponents. This conversion is very useful, as seen in our example, making it easier to apply exponent rules.
It's essential to understand that simplifying radicals often involves expressing them as fractional exponents, enabling easier manipulation. For example, \(\frac{1}{\sqrt{x}}\) becomes \(x^{-1/2}\), giving us a straightforward method to simplify expressions that include radicals.

• Radicals can often be simplified by recognizing perfect squares.
• Rewriting as fractional powers allows use of exponent rules to simplify further.

Getting comfortable with converting and manipulating radicals will aid in both simplification and solving more complex algebraic equations.