Problem 18 - Elementary Algebra

Question

Find each value. Assume the base is not zero. $$ \frac{y^{7}}{y^{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\frac{y^{7}}{y^{3}}$$. Answer: $$y^4$$.

Step 1: Identify the base and exponents in the given expression

The expression we want to simplify is $$\frac{y^{7}}{y^{3}}$$. Here, the base is y, and the exponents are 7 and 3.

Step 2: Apply the quotient rule for exponents

According to the quotient rule for exponents, when we divide two exponential expressions with the same base, we should subtract the exponent of the denominator from the exponent of the numerator. So, we have: $$ y^{7-3} = y^4 $$

Step 3: Write the simplified expression

The simplified expression after applying the quotient rule for exponents is $$y^4$$.

Key Concepts

Simplifying Exponential Expressions

Understanding how to work with exponential expressions is a fundamental aspect of algebra. Simplifying such expressions often involves using specific algebraic rules designed to reduce complexity and make equations more manageable.

An essential tool in this process is the quotient rule for exponents, which states that when you divide two powers with the same base, you subtract the exponent of the denominator from the exponent of the numerator. For example, given the expression \(\frac{y^{7}}{y^{3}}\), simplification using the quotient rule yields \(y^{7-3} = y^4\).

It's important to remember that the base, in this case, 'y', must be nonzero. Why? Because anything raised to the power of zero is 1, and dividing by zero is undefined in mathematics, a situation we always aim to avoid.

Exponents and Powers

Exponents and powers are shorthand for expressing repeated multiplication. When you see an expression like \(y^7\), it tells you to multiply the base 'y' by itself six more times, for a total of seven factors of 'y'.

Interpreting the Exponent

The exponent (in this case, 7) is crucial as it dictates how many times to use the base as a factor. If the exponent were to change, so too would the value of the expression. Negative exponents and fractional exponents introduce additional rules and interpretations, further expanding the utility of exponential notation.

In the context of simplification using the quotient rule, the behavior of exponents allows us to combine and simplify related terms with ease, turning potentially challenging equations into more straightforward expressions that are easier to evaluate or use in subsequent calculations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that collectively represent a specific quantity. In algebra, we frequently encounter expressions incorporating exponents, like \(y^7\) or \(y^3\), especially when dealing with polynomial functions, exponential growth, or decay models.

The ability to manipulate these expressions through rules like the quotient rule for exponents is crucial. It empowers students to solve equations, simplify formulas, and understand more complex functions. The quotient rule not only simplifies individual terms but also helps in reducing entire algebraic expressions down to their simplest forms, enabling a clearer view of the mathematical relationships at play.

Mastering the manipulation of algebraic expressions using exponent rules is key not only for success in algebra but in higher mathematics where such expressions often provide the foundation for more sophisticated concepts and problem-solving strategies.