Problem 45 - Algebra 1: Concepts and Skills

Question

Simplify the quotient. $$ \left(\frac{x}{y}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified form of \(\left(\frac{x}{y}\right)^{5}\) is \(\frac{x^5}{y^5}\).

Step 1: Apply the power of a quotient rule

The power of a quotient rule states \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\).

Step 2: Simplify

\(\left(\frac{x}{y}\right)^5 = \frac{x^5}{y^5}\)

Key Concepts

Simplification of Expressions

Simplification of expressions is a fundamental skill in algebra. It involves reducing expressions to a simpler or more compact form without changing their value. In the given exercise, the expression is a quotient of powers, \( \left(\frac{x}{y}\right)^{5} \). To simplify this expression, we use the power rule for exponents, which is a way to manage expressions that involve powers efficiently.

By applying the power rule in this case, the expression can be broken down into simpler terms. This involves raising both the numerator and the denominator of the fraction to the power of 5 separately. By simplifying this complex expression into \( \frac{x^5}{y^5} \), we make it easier to understand and further manipulate algebraically.

The simplified expression is more straightforward and can be used in further mathematical operations without complication. This process of simplification is crucial as it makes subsequent calculations more manageable and less prone to errors.

Quotients in Algebra

Understanding quotients in algebra is crucial for solving many mathematical problems. A quotient represents the division of one quantity by another. In algebraic terms, a fraction is a form of a quotient where the top number (numerator) is divided by the bottom number (denominator).

In this exercise, the initial expression \( \left(\frac{x}{y}\right)^{5} \) shows a fraction raised to a power. Quotients in such expressions are simplified by applying the power rule, which allows us to distribute the exponent to both the numerator and the denominator separately.

This means each variable in the fraction gets raised to the power individually, resulting in \(x^5 \) divided by \(y^5\). Hence, understanding how to manipulate quotients and apply rules correctly is vital in simplifying algebraic expressions efficiently, making them easier to work with in further equations and calculations.

Exponential Expressions

Exponential expressions involve numbers or variables raised to a power, signifying repeated multiplication. For example, in the expression \( x^5 \), the base \( x \) is multiplied by itself five times.

The task involves simplifying \( \left(\frac{x}{y}\right)^{5} \) which includes applying the power to a fraction. Exponential expressions follow specific rules that help in their simplification. One such rule is the power rule, which states that when a fraction is raised to a power, both the numerator and the denominator are raised to that power individually.

• This results in \( x^5 \) being the numerator.
• Similarly, \( y^5 \) becomes the denominator.

These manipulations allow the expression \( \left(\frac{x}{y}\right)^{5} \) to be turned into \( \frac{x^5}{y^5} \), a much more simplified and usable form for further algebraic operations.

Mastering the concepts of exponential expressions will equip students with the necessary skills to tackle more complex algebraic tasks, including equations and functions that involve various types of exponents.