Problem 2 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \frac{30 y^{4}}{-5 y^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-6y^2\).

Step 1: Simplify the Coefficients

First, look at the coefficients in the expression. We have \(30\) in the numerator and \(-5\) in the denominator. Simplify by dividing both coefficients by 5: \(\frac{30}{-5} = -6\).

Step 2: Simplify the Exponents

Now, focus on the exponents of \(y\). The numerator has \(y^4\) and the denominator has \(y^2\). Use the quotient rule for exponents, \(\frac{y^a}{y^b} = y^{a-b}\): \(y^{4-2} = y^2\).

Step 3: Combine the Results

Combine the simplified coefficient and variable with exponent to get the final simplified expression: \(-6y^2\).

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, we mean making them as concise and easy to work with as possible. In algebra, this often involves reducing fractions, combining like terms, or applying mathematical rules to transform the expression into its simplest form.
To simplify a rational expression, you deal with two main parts:

• The numerical coefficients
• The variable parts with exponents

In the example given, the expression is \(\frac{30 y^{4}}{-5 y^{2}}\). The goal is to simplify the expression step by step. Start with the numerical coefficients by dividing 30 by -5 to get -6. Then, use rules applicable to exponents to simplify the variable part, \(y^{4}\) and \(y^{2}\).
Combining these results leads to the expression being reduced to \(-6y^2\). Always remember that simplifying expressions not only makes them more manageable but also prepares them for solving any further algebraic manipulation or equation solving.

Exponents

Exponents are a way to represent repeated multiplication of the same number or variable. For example, \(y^4\) means \(y \times y \times y \times y\). Understanding how to work with exponents is crucial in algebra because they frequently appear in many different types of expressions and equations.
Exponents have some basic rules:

• Multiplication: \(x^a \times x^b = x^{a+b}\)
• Division (Quotient Rule): \(\frac{x^a}{x^b} = x^{a-b}\)
• Power of zero: \(x^0 = 1\), provided \(x eq 0\)
• Negative exponents: \(x^{-a} = \frac{1}{x^a}\)

In algebraic expressions, knowing how to manipulate exponents allows you to simplify expressions easily and solve equations more effectively. In our exercise, the quotient rule helps to simplify \(y^4\) divided by \(y^2\) to \(y^{4-2}\), resulting in \(y^2\).
Practicing these rules will aid in effortlessly handling complex expressions you may encounter.

Quotient Rule for Exponents

The quotient rule for exponents is a key concept in algebra. It provides a straightforward way to simplify expressions where a base with an exponent is divided by the same base with another exponent.
The rule states: \(\frac{x^a}{x^b} = x^{a-b}\). Here’s how it works in practice:

• Identify the base that is common in both the numerator and the denominator.
• Subtract the exponent in the denominator from the exponent in the numerator.
• Write the result with the common base and the new exponent.

Applying this rule in the given problem \(\frac{y^4}{y^2} = y^{4-2}\), simplifies directly to \(y^2\). This rule is especially useful as it allows complex expressions to be simplified into a much easier form, making subsequent algebraic operations significantly easier.
Learning and practicing this rule will improve your efficiency and accuracy in solving algebraic expressions involving exponents.