Problem 119 - Algebra A Combined Function

Question

Fill in the boxes so that each statement is true. $$ \frac{y^{\square}}{y^{\square}}=y^{7} $$

Step-by-Step Solution

Answer

Fill the boxes with 14 and 7 respectively.

Step 1: Understand the Given Expression

The expression given is \( \frac{y^{\square}}{y^{\square}} = y^{7} \). To solve this, we need to fill in the blanks in the exponents such that the left-hand side simplifies to the expression on the right-hand side.

Step 2: Apply Properties of Exponents

Recall that the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \) allows us to simplify the expression. This means that \( \frac{y^{\square}}{y^{\square}} \) is \( y^{\square - \square} \).

Step 3: Set Up Equation from the Exponents

For the statement \( \frac{y^{\square}}{y^{\square}} = y^{7} \) to be true, the exponents must be equal. This implies \( \square - \square = 7 \).

Step 4: Solve for the Exponents

For \( \square - \square = 7 \) to be true, assume that the top exponent \(a\) and the bottom exponent \(b\) must satisfy \( a - b = 7 \). Any values of \(a\) and \(b\) that satisfy this equation will run true.

Step 5: Select Values for Boxes

A simple solution is choosing \(a = 14\) and \(b = 7\), so that \( 14 - 7 = 7 \). Therefore, the boxes should be filled with these values: \( \frac{y^{14}}{y^{7}} = y^{7} \).

Key Concepts

Exponent Rules

Exponents are a fundamental part of algebra, and understanding their rules is crucial for simplifying expressions and solving problems efficiently. When working with exponents, several key properties can be applied to simplify and solve equations. Two important rules are:

• The Product Rule: When multiplying like bases, add the exponents: \( a^m \times a^n = a^{m+n} \).
• The Quotient Rule: When dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

In the exercise, the rule used is the Quotient Rule of Exponents. This tells us that dividing two expressions with the same base allows their exponents to be subtracted. In our case, \( \frac{y^{\square}}{y^{\square}} \) is simplified to \( y^{\square - \square} \), which should equal \( y^7 \). The goal is to identify values of the exponents in the numerator and denominator that fulfill \( \square - \square = 7 \). Understanding these rules is essential for manipulating and working with powers and bases in algebraic expressions.

Simplifying Expressions

Simplifying expressions involves reducing algebraic expressions to their simplest form. This process often involves employing rules, such as the laws of exponents, to combine and reduce terms effectively.

Consider the expression \( \frac{y^{14}}{y^7} \). Using the quotient rule, we subtract the exponent in the denominator from the exponent in the numerator: \( y^{14-7} \), simplifying to \( y^7 \). This result matches \( y^7 \), confirming the expression is simplified correctly.

• Identify like bases.
• Apply applicable exponent rules (product or quotient rules).
• Reduce or combine terms where possible.
• Check your work to ensure accuracy.

Writing expressions in simplified form not only makes them easier to work with but also leads to clearer solutions in algebraic problem-solving. Mastery of simplifying ensures you handle complex expressions with confidence and ease.

Algebra Problem Solving

Solving algebraic problems involves applying a variety of mathematical techniques to find unknown values. Algebra often requires setting up equations based on given expressions or problems. In this exercise, we're solving for unknown exponents that satisfy the equation \( \frac{y^{\square}}{y^{\square}} = y^7 \).

To solve this, we:

• Recognize the form and pattern of the expression: identify operations and base properties.
• Use exponent rules to set up an equation: apply \( \frac{y^a}{y^b} = y^{a-b} \) to find \( a - b = 7 \).
• Select appropriate values for the variables that make this equation true, such as \( a = 14 \) and \( b = 7 \).

Through algebraic problem-solving, we aim to translate word problems or expressions into solvable mathematical statements. With the correct application of algebra rules, finding solutions becomes a structured and logical process.