Problem 48 - Algebra 1

Question

Simplify the expression. The simplified expression should have no negative exponents. $$ \left(\frac{2 x y^{-2} y^{4}}{3 x^{-1} y}\right)^{-2} \cdot\left(\frac{4 x y}{2 x^{-1} y^{-3}}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression with no negative exponents is \(\frac{1}{9}\).

Step 1: Distribute the exponent in each parenthesis

Rewrite the expression, applying the negative two exponent to each term inside the first parenthesis and the positive two exponent to each term inside the second parenthesis: \\\[\left(\frac{2^{-2} x^{-2} y^{2} y^{-8}}{3^{-2} x^{2} y^{-2}}\right) \cdot \left(\frac{4^{2} x^{2} y^{2}}{2^{2} x^{2} y^{6}}\right) \]

Step 2: Simplify the expressions

Multiply all terms with the same base, keep in mind that when multiplying terms with the same base, you add the exponents. Applying that to the terms we have: \\\[\left(\frac{2^{-2} \cdot x^{-2} \cdot y^{-6}}{3^{-2} \cdot x^{2} \cdot y^{-2}}\right) \cdot \left(\frac{4^{2} \cdot x^{2} \cdot y^{-4}}{2^{2} \cdot x^{2}}\right) \]

Step 3: Perform the exponent arithmetic.

In this step, simplify the expressions involving exponents: \\\[\left(\frac{1/4 \cdot 1/x^{2} \cdot 1/y^{6}}{1/9 \cdot x^{2} \cdot 1/y^{2}}\right) \cdot \left(\frac{16 \cdot x^{2} \cdot 1/y^{4}}{4 \cdot x^{2}}\right) \]

Step 4: Simplify fractions

Now, simplify the fractions within each parenthesis and simplify the whole expression: \\\[\left(\frac{1}{36}\right) \cdot \left(4\right) = \frac{1}{9}\]

Key Concepts

Understanding Negative Exponents

Working with negative exponents can be puzzling at first, but it is a fundamental concept in algebra that simplifies expressions while retaining their values. An expression with a negative exponent represents the reciprocal of the base raised to the opposite positive exponent. For example, the expression \( a^{-n} \) is equivalent to \( \frac{1}{a^{n}} \), where \( a \) is the base and \( n \) is the exponent. This change from a negative exponent to a positive one is essential when simplifying algebraic expressions.

When facing an exercise like the textbook example, the key is to apply this rule to each term with a negative exponent before attempting further simplifications. This allows us to transform terms like \( y^{-2} \) into \( \frac{1}{y^{2}} \), making it much simpler to manipulate and combine terms later on.

Navigating Exponent Arithmetic

Mastering exponent arithmetic is crucial for simplifying algebraic expressions efficiently. The rules of exponents outline how to multiply, divide, and raise powers to powers. When multiplying terms with the same base, as shown in our example exercise, we add the exponents, translating \( x^{m} \cdot x^{n} = x^{m+n} \). Similarly, dividing terms with the same base involves subtracting exponents, so \( \frac{x^{m}}{x^{n}} = x^{m-n} \).

When an entire expression is raised to an exponent, every term inside the parentheses is affected by that exponent. This is what we do in the exercise when distributing the exponent of \( -2 \) across the terms within the first set of parentheses and the exponent of \( 2 \) within the second set. Careful manipulation of these rules ensures that even complex expressions with multiple terms and exponents can be simplified to a more manageable form.

Simplify Fractions in Algebra

When simplifying fractions in algebra, the goal is to reduce the expression to its simplest form. This process often involves converting negative exponents to positive ones, as we've covered, and then reducing fractions by canceling common factors in the numerator and denominator.

In our textbook example, we transform the given complex expression into a series of more straightforward fractions, which are then simplified by canceling out common terms. It's important to remember that when a term appears in both the numerator and the denominator, they can often be simplified, much like reducing \( \frac{4}{8} \) to \( \frac{1}{2} \) in traditional arithmetic. By carefully simplifying each fraction within the terms of an algebraic expression, we can drastically simplify the whole algebraic expression into something like \( \frac{1}{9} \), a much simpler form to work with.