Problem 62 - Algebra for College Students

Question

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(y^{\frac{3}{4}}\right)\left(y^{-\frac{1}{2}}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \( y^{\frac{1}{4}} \).

Step 1: Identify the Problem

We need to simplify the expression \( \left(y^{\frac{3}{4}}\right)\left(y^{-\frac{1}{2}}\right) \) and express the result using only positive exponents.

Step 2: Applying Exponent Rules

Utilize the product of powers rule, which states that \( a^m \times a^n = a^{m+n} \). Applying this rule to the expression \( y^{\frac{3}{4}} \times y^{-\frac{1}{2}} \), we add the exponents: \( \frac{3}{4} + \left(-\frac{1}{2}\right) \).

Step 3: Simplify the Exponent

Perform the addition of the exponents: \( \frac{3}{4} - \frac{1}{2} \). To do this, convert \( \frac{1}{2} \) to a fraction with the same denominator as \( \frac{3}{4} \). This becomes \( \frac{2}{4} \), so we have \( \frac{3}{4} - \frac{2}{4} = \frac{1}{4} \).

Step 4: Write the Simplified Expression

The expression simplifies to \( y^{\frac{1}{4}} \) since \( y^{\frac{3}{4} - \frac{1}{2}} = y^{\frac{1}{4}} \). This expression uses only positive exponents.

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing an algebraic expression to its most concise and easy-to-use form. This is often done by following certain rules and mathematical operations.
For instance, combining like terms, applying exponent rules, and simplifying fractions as in the example \( \left(y^{\frac{3}{4}}\right)\left(y^{-\frac{1}{2}}\right) \) in the original exercise.

• Identify like terms and combine them.
• Use mathematical properties to combine coefficients and variables.
• Simplify the exponents by doing the arithmetic on the powers.

Following these steps helps us to work with simpler forms of expressions, making further calculations easier and clearer.

Positive Exponents

When simplifying expressions, it is often required to express the final result using only positive exponents. Positive exponents represent a straightforward multiplication of a base number, while negative exponents indicate the reciprocal.
For example, \( y^{-\frac{1}{2}} \) can be rewritten as \( \frac{1}{y^{\frac{1}{2}}} \).

• To convert a negative exponent to positive, find the reciprocal of the base raised to the corresponding positive power.
• Use positive exponents to improve readability and understanding of the expression.
• Remember that this is a fundamental practice in mathematics, especially in algebra.

Maintaining positive exponents in final expressions ensures that there are no fractions in the equation related to exponentiation directly, ensuring clarity and ease of further calculations.

Product of Powers Rule

The product of powers rule is an essential exponent rule in mathematics. It states that when multiplying two exponential expressions with the same base, you simply add their exponents.
In the expression \( y^{\frac{3}{4}} \times y^{-\frac{1}{2}} \), both parts share the base \( y \). According to the rule, you add the exponents: \( \frac{3}{4} + \left(-\frac{1}{2}\right) = \frac{1}{4} \).

• Make sure the bases of the terms are identical before applying this rule.
• Add the exponents together, paying careful attention to the signs.
• Ensure the final expression has positive exponents if required.

The product of powers rule simplifies multiplication in exponential expressions, making it easier to handle complex calculations, especially those involving fractions and negative exponents.