Problem 78 - Algebra and Trigonometry

Question

Perform the indicated operations, and simplify. \(x^{1 / 4}\left(2 x^{3 / 4}-x^{1 / 4}\right)\)

Step-by-Step Solution

Answer

The simplified form is \(2x - x^{1/2}\).

Step 1: Distribute the Term

Distribute the term \(x^{1/4}\) across the expression in the parentheses. This means you will multiply \(x^{1/4}\) by each term inside: \[x^{1/4} \cdot (2x^{3/4}) - x^{1/4} \cdot (x^{1/4})\]

Step 2: Simplify the First Term

Multiply \(x^{1/4}\) by \(2x^{3/4}\). Since we can multiply the coefficients and add the exponents, we have:\[2 \cdot x^{1/4+3/4} = 2x\]

Step 3: Simplify the Second Term

Multiply \(x^{1/4}\) by \(x^{1/4}\). This involves adding the exponents:\[x^{1/4+1/4} = x^{1/2}\]

Step 4: Combine the Simplified Terms

Taking the results from steps 2 and 3, we can combine the simplified terms:\[2x - x^{1/2}\]

Key Concepts

Operations with Exponents

When dealing with exponents, there are several rules that can help simplify calculations. An exponent indicates how many times you multiply a number by itself. Key operations include:

• Multiplying with the same base: Add the exponents. For example, if you have \(a^m \cdot a^n\), the result is \(a^{m+n}\).
• Dividing with the same base: Subtract the exponents. In the case of \(a^m / a^n\), this simplifies to \(a^{m-n}\).
• Power of a power: Multiply the exponents. For \((a^m)^n\), it becomes \(a^{mn}\).

In the original exercise, multiplying \(x^{1/4}\) by other exponents involved simply adding the exponents together. This is an essential operation to learn and practice, as it appears frequently in algebraic problems. Always remember, working with exponents is like working with the counters of multiplication operations.

Simplifying Expressions

Simplifying an algebraic expression means transforming it into its simplest form. The goal is to make it easier to understand or work with. This often involves combining like terms and applying algebraic rules.

• Combine like terms: Terms that have the same variable raised to the same power can be added or subtracted. If you have \(3x + 2x\), it simplifies to \(5x\).
• Apply exponent rules: As discussed, use rules like the addition or subtraction of exponents to simplify expressions.
• Break it down: Look at complex expressions as collections of simpler expressions and deal with one part at a time.

In the solution provided, after distributing and simplifying each term by combining like terms and using exponent operations, the expression was narrowed down to \(2x - x^{1/2}\). This made it simpler and clearer.

Distributive Property

The distributive property is a key concept in algebra that involves distributing or spreading terms across parentheses. It states that when a term is multiplied by terms within parentheses, it should multiply each of those terms individually. Mathematically, this is expressed as:

• \(a(b+c) = ab + ac\)
• In reverse, \(ab + ac = a(b+c)\)

This property is particularly useful for simplifying expressions and solving equations. In the given exercise, the distributive property was used to multiply \(x^{1/4}\) with each term inside the parentheses. This crucial step set the stage for simplifying the entire expression. It's like unpacking a box; each item needs attention to properly organize or solve the bigger problem.