Problem 54 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \frac{y^{4}}{y^{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression and state the final result: $$\frac{y^{4}}{y^{3}}$$ Answer: The simplified expression is: $$\frac{y^{4}}{y^{3}} = y$$

Step 1: Identify the base and exponents

We are given the expression: $$ \frac{y^{4}}{y^{3}} $$ Here, the base is \(y\), and the exponents are \(4\) and \(3\).

Step 2: Apply the quotient rule of exponents

According to the quotient rule of exponents, when we divide two expressions with the same base, we simply subtract the exponent of the denominator from the exponent of the numerator. So, we have: $$ \frac{y^{4}}{y^{3}} = y^{(4-3)} $$

Step 3: Simplify the expression

Now, we will subtract the exponents to simplify the expression: $$ y^{(4-3)} = y^1 $$ Since the exponent is \(1\), we can further simplify the expression to: $$ y^1 = y $$ Therefore, the simplified expression is: $$ \frac{y^{4}}{y^{3}} = y $$

Key Concepts

Product Rule of Exponents

Mastering the product rule of exponents is crucial for simplifying algebraic expressions efficiently. The product rule states that when you multiply two expressions with the same base, you keep the base and add the exponents. Consider the example \( x^a \cdot x^b = x^{a+b} \). This is particularly handy when tackling complex problems involving multiplication of variables with exponents.

Understanding and applying the product rule can make algebraic operations less daunting. When faced with expressions like \( x^3 \cdot x^2 \), remember that you simply need to add the exponents due to the common base \(x\), resulting in \( x^{3+2} = x^5 \). Such simplification is foundational before approaching exercises with more advanced concepts.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that helps make equations more manageable. The goal is to reduce an expression to its simplest form by combining like terms and using algebraic rules. It's beneficial to follow a systematic process: combine like terms, apply the product and quotient rules of exponents, and carry out any addition or subtraction.

For instance, an expression like \( 3x^2 + 2x^2 \) can be simplified by combining the like terms to get \( 5x^2 \). Simplification not only makes the numbers easier to work with but also helps in understanding the structure of algebraic equations better. Students are advised to always look for opportunities to streamline expressions for clarity and ease of use.

Algebraic Exponents

Algebraic exponents are exponents that appear in algebra and are used to represent repeated multiplication. They are a convenient way of expressing large numbers or expressions in a compact form. A number like \( a \), raised to the power of \( n \), written as \( a^n \), means \( a \times a \times ... \times a \) (where \( a \) is multiplied by itself \( n \) times).

There are important rules associated with exponents that the students must memorize, such as the product rule, quotient rule, power of a power rule, and negative exponent rule, each of which has specific applications. For example, \( (a^m)^n = a^{mn} \), which is the power of a power rule, suggests that when an exponential expression is raised to another exponent, you multiply the exponents. These rules are the building blocks for working with algebraic expressions at all levels of complexity.