Problem 122 - Intermediate Algebra

Question

Fill in each box with the correct expression. $$ \frac{\square}{y^{-3 / 4}}=y^{4 / 4}, \text { or } y $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Given Expression

We are given the expression \( \frac{\square}{y^{-3/4}} = y \). Our task is to find the expression that should fill the box, represented by \( \square \).

Step 2: Express \( y \) in Terms of the Denominator Exponent

The right side of the equation is \( y \), which can be expressed in exponential form as \( y^{1} \).

Step 3: Multiply Both Sides by \( y^{-3/4} \)

To find \( \square \), multiply both sides of the equation by \( y^{-3/4} \), to isolate \( \square \) on one side:\[ \square = y^{1} \times y^{-3/4} \]

Step 4: Apply Exponent Rules

Use the property of exponents, \( a^m \times a^n = a^{m+n} \), to simplify the expression:\[ y^{1} \times y^{-3/4} = y^{1 - 3/4} \] which simplifies to \( y^{1/4} \).

Step 5: Write the Final Expression

The expression that fills the box is \( y^{1/4} \). Thus, the complete equation is:\[ \frac{y^{1/4}}{y^{-3/4}} = y \]

Key Concepts

Understanding Exponents

Exponents play a crucial role in mathematics, simplifying expressions and enabling the compact representation of repeated multiplication. An exponent refers to the number of times a base number is multiplied by itself. For example, in the expression \( x^n \), \( x \) is the base while \( n \) is the exponent. Here's how exponents work in different scenarios:

• If \( n \geq 0 \), \( x^n \) means multiplying \( x \) by itself \( n \) times.
• If \( n = 0 \), \( x^0 = 1 \), for any non-zero \( x \).
• If \( n < 0 \), \( x^n \) is the reciprocal of \( x^{-n} \), meaning \( \frac{1}{x^{-n}} \).

When dealing with a rational expression, such as in the exercise, remember to handle the exponents carefully as they guide the manipulation of terms. For instance, converting an exponent from the denominator to the numerator requires flipping its sign. This is known as the negative exponent property. Understanding and applying these rules is essential when working with equations involving irrational or fractional exponents.

Mastering Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to isolate variables or reach a desired form. It's an essential skill in algebra that allows mathematicians to solve complex equations more easily.

There are fundamental techniques to keep in mind:

• Collect Like Terms: Combine terms with the same variable parts. For example, \( 3x + 2x = 5x \).
• Use Exponent Laws: When multiplying terms with the same base, you can add the exponents: \( a^m \times a^n = a^{m+n} \).
• Distribute and Factor: Use the distributive property to expand expressions and factor them to simplify.

In the given exercise, multiplying both sides by the denominator's exponent allowed us to eliminate the fraction and focus on the remaining simplification. Practicing these manipulation skills is key to mastering algebra and solving equations efficiently.

Solving Equations with Rational Expressions

Solving equations that involve rational expressions requires a solid understanding of both the nature of fractions and exponential rules. In the exercise, we started with a fractional expression and aimed to find the missing value by manipulating the equation. Here’s how to approach these problems:

• Understand the Equation: Start by clearly understanding what each part of the equation represents.
• Clear the Fraction: Multiply both sides by the expression in the denominator to eliminate the fraction. In the example, multiplying by \( y^{-3/4} \) isolated our missing term.
• Simplify with Exponents: Use exponent rules to simplify the resulting expression. In this case, adding the exponents \( 1 \) and \(-3/4\) yielded the simplified form \( y^{1/4} \).

Approach each step methodically, and always double-check your simplifications to ensure accuracy. By following these steps, you can effectively solve equations involving rational expressions, yielding the correct answer reliably.