Problem 61 - Algebra 1

Question

Simplify the fraction. $$\frac{y^{4} \cdot y^{7}}{y^{5}}$$

Step-by-Step Solution

Answer

The simplified version of the fraction \(\frac{y^{4} \cdot y^{7}}{y^{5}}\) is \(y^{6}\).

Step 1: Simplify Numerator

Two terms are being multiplied in the numerator, both terms having the same base 'y'. When multiplying terms with the same base, the exponents are added. Therefore, \(y^{4} \cdot y^{7}\) simplifies to \(y^{4+7}\), which is \(y^{11}\).

Step 2: Apply Division Rule

The operation between the numerator \(y^{11}\) and the denominator \(y^{5}\) is division. When dividing two terms with the same base, their exponents are subtracted. Hence, \(y^{11} ÷ y^{5}\) simplifies to \(y^{11-5}\), which is \(y^{6}\).

Key Concepts

Exponent Rules

Understanding how to manipulate exponents is vital for simplifying algebraic fractions. Exponents, or powers, are a shorthand way to indicate that a number or variable is multiplied by itself a specific number of times. For instance, in our exercise, we have the term \(y^{4}\) which means y is multiplied by itself 4 times.

In algebra, certain rules make working with exponents easier. One of these rules is the Product of Powers rule, which states that when multiplying two expressions that have the same base, you can add the exponents. So, \(y^{4} \times y^{7} = y^{4+7} = y^{11}\).

Another rule is the Quotient of Powers, used when dividing expressions with the same base. According to this rule, you subtract the exponent of the denominator from the exponent of the numerator. That’s how \(y^{11} ÷ y^{5}\) becomes \(y^{11-5} = y^{6}\). These exponent rules are the backbone of simplifying many algebraic expressions.

Simplifying Expressions

Simplifying expressions is the process of reducing complexity in an algebraic expression, making it easier to work with. In the case of our exercise, the simplification process starts by applying exponent rules. However, it doesn't end there. Simplification can also include combining like terms, removing parentheses, or even factoring.

To master simplification, practice recognizing patterns and applying appropriate algebraic rules. For example, if ever faced with complex fractions or additional terms, look for common factors or terms that can cancel each other out. Keep in mind that the goal is to express the equation in its simplest form without changing its value.

Algebraic Operations

Algebraic operations, inclusive of addition, subtraction, multiplication, and division, form the foundation for manipulating expressions. In our exercise, we used both multiplication and division. But algebraic operations also include factoring, expanding polynomials, and working with inequalities, among others.

Each operation follows specific rules to ensure the integrity of the expressions is maintained. For instance, when multiplying polynomials, you must distribute each term accordingly, known as the distributive property. Understanding these operations in depth empowers students to tackle various algebraic problems with confidence. As you become more practiced with these operations, working with algebraic fractions and more complicated expressions will become significantly easier.