Problem 54 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. \(\left(6^{2}-24\right)^{2} \div\left(\frac{x}{y}\right)^{-5}\)

Step-by-Step Solution

Answer

The simplified expression is \(144 \frac{x^5}{y^5}\).

Step 1: Simplify the Expression Inside the Parentheses

First, evaluate the expression inside the parentheses: \(6^2 - 24\). Calculate \(6^2\), which is 36. Then, subtract 24 from 36: \(36 - 24 = 12\). The expression simplifies to \(12\).

Step 2: Square the Simplified Expression

Next, take the result from Step 1, which is 12, and square it: \(12^2 = 144\).

Step 3: Invert the Fraction with a Negative Exponent

The expression involves dividing by \(\left(\frac{x}{y}\right)^{-5}\). To simplify this part, understand that a negative exponent indicates taking the reciprocal. Thus, \(\left(\frac{x}{y}\right)^{-5} = \left(\frac{y}{x}\right)^5\).

Step 4: Divide the Squared Result by the Reciprocal Expression

To complete the expression, divide the result from Step 2 by the result from Step 3. This gives us: \[ 144 \div \left(\frac{y}{x}\right)^5 = 144 \times \left(\frac{x}{y}\right)^5 = 144 \times \left(\frac{x^5}{y^5}\right) \].

Step 5: Write the Final Simplified Expression

Combining all the steps, the expression is simplified to: \[ 144 \frac{x^5}{y^5} \]. This form uses only positive exponents.

Key Concepts

Exponents and Powers

Exponents and powers are fundamental concepts when it comes to simplifying algebraic expressions. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \(6^2\), 6 is the base, and 2 is the exponent, meaning you multiply 6 by itself, i.e., \(6 \times 6 = 36\).

• An integer exponent, like 2 in \(6^2\), suggests repeated multiplication of the base.
• Powers are simply the result of this multiplication process, in this case, the number 36.

Understanding exponents helps quickly calculate values and simplify expressions.
When simplifying, follow the exponentiation rule: base raised to an exponent is multiplied by itself as many times as the value of the exponent.

Negative Exponents

Negative exponents can initially seem intimidating, but they actually simplify our calculations by introducing the concept of reciprocation. A negative exponent means you take the reciprocal of the base to the opposite positive exponent. For instance, \(x^{-2}\) is equivalent to \(\frac{1}{x^2}\).
This principle is critical in simplifying expressions with negative exponents.

• Moving from negative to positive exponents simplifies handling complex divisions.
• Negative exponents indicate reciprocal fractions, which are particularly useful when dividing terms.

By converting negative exponents to positive ones using reciprocals, expressions become clearer and easier to manage.

Fractional Expressions

Fractional expressions often involve terms with both numerators and denominators that include variables or numbers. Simplifying these requires a clear understanding of fraction properties and exponent rules. In algebraic terms, expressions like \(\frac{x}{y}\) can be raised to a power, possibly involving negative exponents.
Converting negative exponents to fractions is crucial, as seen in \(\left(\frac{x}{y}\right)^{-5}\), which becomes \(\left(\frac{y}{x}\right)^5\).

• Simplify fractional expressions by applying the power uniformly to both numerator and denominator.
• Recognize how negative exponents affect the position of numbers or variables within fractions.

This helps align fractions with straightforward multiplication or division processes in calculations.

Order of Operations

To reliably simplify expressions, follow the order of operations. This ensures accurate computation and simplification. The standard order of operations can be remembered using PEMDAS:

• P: Parentheses should be simplified first.
• E: Exponents and powers come next.
• M/D: Multiplication and Division are tackled together, from left to right.
• A/S: Addition and Subtraction complete the order, from left to right.

In dealing with the exercise, simplifying inside the parentheses first helps resolve the inherent calculations correctly. Next, tackle exponents before engaging the multiplication or division operations. Maintaining this disciplined approach helps in breaking down complex algebraic expressions into manageable parts, resulting in a simplified output.