Problem 51 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponents \((\mathrm{s}) .\) Assume that all letters denote positive numbers. \(\left(y^{3 / 4}\right)^{2 / 3}\)

Step-by-Step Solution

Answer

The simplified expression is \(y^{1/2}\).

Step 1: Understand the Problem

We need to simplify the given expression \(\left(y^{3 / 4}\right)^{2 / 3}\) and ensure no negative exponents remain.

Step 2: Apply the Power of a Power Property

When raising a power to a power, the exponents should be multiplied. The expression \(\left(y^{3 / 4}\right)^{2 / 3}\) can be simplified by multiplying the exponents \(\frac{3}{4}\) and \(\frac{2}{3}\).

Step 3: Multiply the Exponents

Calculate \(\frac{3}{4} \times \frac{2}{3} = \frac{6}{12}\). Simplify this fraction to \(\frac{1}{2}\).

Step 4: Rewrite the Expression

Rewrite the expression with the new exponent: \(y^{1/2}\).

Step 5: Final Expression

The simplified expression is \(y^{1/2}\). Since \(y\) is assumed to be positive, there are no negative exponents to worry about.

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making a mathematical statement easier to understand or work with. In algebra, this often involves reducing the expression to its simplest form. Basically, you aim to rewrite it so it's less complex but still equivalent to the original.

This process may involve:

• Combining like terms
• Reducing fractions
• Applying mathematical properties (like distributive, associative, commutative)

When you simplify by eliminating negative exponents, it may mean converting them into positive ones. The goal is a "clean" expression that is easily evaluated or graphed. Understanding this basic purpose makes working with algebraic expressions much more straightforward.

Power of a Power Property

The power of a power property is a useful tool when simplifying expressions involving exponents. It states that when you have an exponentiated term raised to another power, the exponents are multiplied. This is expressed by the formula: \[(a^m)^n = a^{mn} \]Applying this property helps quickly handle expressions that might seem complex initially. Let's take \((y^{3/4})^{2/3} \)as an example once more. Following our property, multiply the exponents: \( \frac{3}{4} imes \frac{2}{3} = \frac{6}{12} \).Simplifying \(\frac{6}{12} \)yields \(\frac{1}{2} \).Now the expression becomes \( y^{1/2} \), making it simpler to manage. This makes exponent handling much more organized.

Positive Numbers

Working with positive numbers in algebra provides some significant simplifications. Positive numbers are those larger than zero and they play nicely with exponents, making calculations straightforward.

When solving problems or simplifying expressions, assuming variables represent positive numbers can prevent issues like undefined expressions (which can occur with negative bases raised to non-integer exponents).

• Positive numbers simplify calculations
• Keep solutions in real numbers, avoiding complex numbers

If you know variables, like \(y\),in an expression \(y^{1/2}, \) represent positive numbers, it reassures us that the steps followed lead to valid and meaningful algebraic manipulation.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and operational symbols. They are the backbone of algebraic calculations. Understanding them is critical to working with algebra efficiently and effectively.

An expression can take forms such as:

• Simple, like \(x + 3 \)
• Complex with multiple operations, like \((2x^2 - 3y + 7) \)

In simplifying an expression like \( \left(y^{3/4}\right)^{2/3} \), you're essentially transforming it into a simpler form \(y^{1/2} \). You maintain equivalence while ensuring it’s easier to comprehend and use. Having strong fundamental skills in interpreting and manipulating these expressions allows for successful problem-solving in algebra.