Problem 90 - College Algebra with Modeling and Visualization

Question

Exercises \(65-90:\) Use rules of exponents to simplify the expression. Use positive exponents to write your answer. $$ \left(\frac{4(x y)^{2}}{\left(2 x y^{-2}\right)^{3}}\right)^{-2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{4x^{2}}{y^{16}}\).

Step 1: Simplify the Denominator

First, simplify the expression in the denominator: \((2xy^{-2})^{3}\). Apply the exponent to each factor inside the parentheses:\[(2xy^{-2})^{3} = 2^{3} imes x^{3} imes (y^{-2})^{3} = 8x^{3}y^{-6}\].

Step 2: Substitute Back into the Fraction

Replace the denominator in the original expression with the simplified form from Step 1:\[\left(\frac{4(xy)^{2}}{8x^{3}y^{-6}}\right)^{-2}\].

Step 3: Apply the Negative Exponent to the Fraction

The negative exponent \(-2\) means to take the reciprocal of the fraction and then apply the exponent:\[\left(\frac{8x^{3}y^{-6}}{4(xy)^{2}}\right)^{2}\].

Step 4: Simplify the New Fraction

Simplify the fraction before applying the exponent:\[\frac{8x^{3}y^{-6}}{4x^{2}y^{2}} = 2x^{1-2}y^{-6-2} = 2x^{1}y^{-8}\].

Step 5: Apply the Exponent to Each Term

Now apply the exponent of \(2\) to each factor in the simplified fraction:\[(2x^{1}y^{-8})^{2} = 2^{2}x^{2}(y^{-8})^{2} = 4x^{2}y^{-16}\].

Step 6: Ensure Positive Exponents

Convert any negative exponents to positive by rewriting fractions:\[4x^{2}\frac{1}{y^{16}} = \frac{4x^{2}}{y^{16}}\].

Key Concepts

Understanding Algebraic Expressions

Algebra is a foundational branch of mathematics that deals with symbols and the rules for manipulating these symbols. In the provided exercise, we work with expressions that involve variables like \(x\) and \(y\), along with constants such as numbers like 4 and 2.

An algebraic expression can be considered as a mathematical phrase that combines numbers, variables, and operational symbols. Understanding how to manage and simplify these expressions is crucial in algebra.

In the given problem, we see expressions raised to powers, both positive and negative, requiring us to apply specific algebraic rules. Simplifying expressions is a principal task here, and it often involves getting rid of fractions or negative exponents to express the result in the simplest, most understandable way.

Working through algebra problems requires patience. Ensuring that each step logically follows the last helps in unraveling even the most complex expressions.

Simplification in Algebra

Simplification is a critical skill in algebra, focusing on reducing expressions to their simplest form. In this example, the complex fraction needs simplification by clearing out negative exponents and reducing coefficients.

Key steps in simplification involve:

• Breaking down each term within an expression to its base components.
• Applying exponent rules to merge terms or adjust exponents.
• Ensuring the final result has no fractions or negative powers for an optimal presentation.

The rule of reciprocals is used when handling negative exponents, meaning you "flip" the fraction to make the exponent positive. By simplifying the expression, you convert an initially confusing jumble into a clear solution, saving time and effort later down the problem-solving path.

Dealing with Negative Exponents

Negative exponents can often seem intimidating, but they are simply another way to represent division. In algebra, a negative exponent signifies the reciprocal of the base raised to the corresponding positive power.

For example, \(a^{-n} = \frac{1}{a^n}\). This means you invert the base and then apply the positive exponent, streamlining expressions by eliminating the negative exponent sign.

The exercise called for the handling of negative exponents, seen in the reciprocal step and in dealing with \(y^{-8}\), which needed conversion to ensure all exponents were positive.

Handling these exponents correctly enables clearer and more direct calculations. It's all about flipping the equation and moving forward with positive values, presenting a neater, final answer.