Problem 16 - Algebra 1: Concepts and Skills

Question

Use the power of a quotient property to simplify the expression. $$ \left(\frac{x}{y}\right)^{6} $$

Step-by-Step Solution

Answer

\((\frac{x}{y})^{6} = \frac{x^{6}}{y^{6}}\).

Step 1: Identify the Expression

We need to evaluate or simplify: \(\left(\frac{x}{y}\right)^{6}\).

Step 2: Apply Order of Operations

Following the order of operations (PEMDAS/BODMAS), we simplify the expression step by step, handling parentheses, exponents, multiplication/division, then addition/subtraction.

Step 3: Compute the Result

Carrying out the operations systematically, we arrive at the result.

Step 4: State the Result

\((\frac{x}{y})^{6} = \frac{x^{6}}{y^{6}}\).

Key Concepts

Understanding Exponents

Exponents are a fundamental concept in mathematics that describe how many times a number, known as the base, is multiplied by itself. In the expression \(a^n\), \(a\) is the base, and \(n\) is the exponent. The expression tells us to multiply \(a\) by itself \(n\) times.

For example, \(2^3 = 2 \times 2 \times 2 = 8\). This represents 2 raised to the power of 3. Understanding exponents is crucial because it allows us to express and calculate large numbers compactly.

In our original problem, we see \((x/y)^6\). Here, the exponent 6 tells us to multiply \(x/y\) by itself 6 times. However, instead of doing this multiplication step by step, we can simplify the expression using exponent rules to make the calculation easier.

Simplifying Expressions with Powers

Simplifying expressions involves reducing them to their simplest form while maintaining their value. When expressions include exponents, we use several rules to simplify them. One such rule is the power of a quotient property.

• Power of a quotient property: This rule states that when you have a fraction raised to an exponent, the exponent applies to both the numerator and the denominator individually. In mathematical terms: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\).
• Example: For the expression \(\left(\frac{x}{y}\right)^6\), we apply the exponent 6 to both \(x\) and \(y\), giving us \(\frac{x^6}{y^6}\).

By applying this property, expressions become more straightforward and easier to handle, especially when solving algebraic equations.

Algebraic Fractions and Simplification

Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions. Being able to simplify these fractions is essential for solving equations, understanding functions, and performing other algebraic operations.

Just like numerical fractions, algebraic fractions can also be simplified. Simplifying involves ensuring that the expression is in its simplest form and is often achieved by applying the power of a quotient property, factoring expressions, or canceling out common terms.

In the context of our problem, the expression \(\left(\frac{x}{y}\right)^6\) was an algebraic fraction raised to an exponent. By applying the power of a quotient property, we simplified it to \(\frac{x^6}{y^6}\). This transformation not only makes it easier to understand and work with the expression but also prepares us for further algebraic manipulations if needed.

Once simplified, if x and y have any common factors that can be canceled or if they represent specific values, further simplification may be possible.