Problem 26 - 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: Distribute the terms

We need to distribute the term \(x^{3/2}\) over the expression inside the parentheses \((\sqrt{x} - \frac{1}{\sqrt{x}})\). This means we will multiply \(x^{3/2}\) by each term inside the parentheses.

Step 2: Multiply \(x^{3/2}\) by \(\sqrt{x}\)

Recall that \(\sqrt{x} = x^{1/2}\). So the operation becomes \(x^{3/2} \cdot x^{1/2}\). By adding the exponents (as they have the same base \(x\)), we get \(x^{(3/2 + 1/2)} = x^{4/2} = x^2\).

Step 3: Multiply \(x^{3/2}\) by \(-\frac{1}{\sqrt{x}}\)

Remember that \(\frac{1}{\sqrt{x}} = x^{-1/2}\). So the operation becomes \(-x^{3/2} \cdot x^{-1/2}\). By adding the exponents, \(3/2 + (-1/2)\), we get \(x^{(2/2)} = x^1 = x\). Therefore the result is \(-x\).

Step 4: Combine the two results

Now, combine the results from Steps 2 and 3: \(x^2 - x\). This is the simplified form of the expression after performing the operations.

Key Concepts

Exponent Rules

When working with algebraic expressions, one of the powerful tools at your disposal is understanding exponent rules. Exponents are a way to represent repeated multiplication of a number by itself. Here, we have specific rules we need to follow:

• Product of Powers Rule: When you multiply terms with the same base, you add their exponents. This means if you have \(a^m \cdot a^n\), it simplifies to \(a^{m+n}\).
• Negative Exponent Rule: An expression with a negative exponent, \(a^{-n}\), is equivalent to the reciprocal of the positive exponent, or \(\frac{1}{a^n}\).
• Exponential Identity Rule: Any number raised to the power of zero is 1, i.e., \(a^0 = 1\).

In our exercise, these rules help us combine the terms \(x^{3/2} \cdot x^{1/2}\) into \(x^2\) by adding the exponents (3/2 + 1/2). Similarly, \(x^{3/2} \cdot x^{-1/2}\) simplifies to \(x\) by adding 3/2 and \(-1/2\), resulting in \(x^1\). Recognizing and applying these rules streamlines the simplification process.

Simplifying Expressions

Simplifying expressions in algebra involves combining like terms and reducing the expression to its simplest form. It transforms a potentially complex-looking equation into something much more understandable. This process often involves:

• Combining Like Terms: These are terms with the same variables raised to the same power. Add or subtract coefficients while keeping the variable parts identical.
• Using Exponent Rules: As mentioned earlier, this includes applying rules for multiplying bases and handling negative exponents.
• Performing Arithmetic Operations: Carry out straightforward arithmetic operations (addition, subtraction, multiplication, division) as needed.

For our original problem, after applying the exponent rules, we simplified \(x^{3/2}(\sqrt{x} - \frac{1}{\sqrt{x}})\) to \(x^2 - x\). Initially, each part of the multiplication was executed separately, using exponent rules, and the results were then combined to provide a simpler form, making the expression less complex and clearer.

Distributive Property

The distributive property is a fundamental truth in mathematics that allows us to distribute multiplication over addition or subtraction inside parentheses. It states that for any numbers or expressions \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equivalent to \(ab + ac\). This means each term inside the parentheses is multiplied by the term outside.

• This property helps break down expressions into smaller, more manageable pieces.
• It is essential when simplifying expressions, particularly when dealing with polynomials and other algebraic structures.

In the problem we tackled, we used the distributive property to multiply \(x^{3/2}\) by each term inside the parentheses \((\sqrt{x} - \frac{1}{\sqrt{x}})\). By distributing \(x^{3/2}\) across the subtraction inside the parentheses, we opened up the expression for simplification by the exponent rules, ultimately simplifying it to \(x^2 - x\). Understanding and applying the distributive property correctly allows more effective handling of algebraic expressions.