Problem 53 - Algebra 1: Concepts and Skills

Question

Simplify the expression. Use only positive exponents. $$ \frac{6 x^{2} y^{2}}{x y^{3}} \cdot \frac{\left(4 x^{2} y\right)^{2}}{x y^{2}} $$

Step-by-Step Solution

Answer

The simplified expression of the given complex fraction, using only positive exponents, is \(96 x^{8}\).

Step 1: Simplify Using Product-to-Power rule

The product-to-power rule states that \((ab)^n = a^n b^n\). So, the expression \( \left(4 x^{2} y\right)^{2}\) can be transformed into \(16 x^{4} y^{2}\). The full expression then becomes: \[\frac{6 x^{2} y^{2}}{x y^{3}} \cdot \frac{16 x^{4} y^{2}}{x y^{2}}.\]

Step 2: Simplify Fractions

Next, simplifying the fractions that involve exponents of the same basis: \[\frac{6 x^{2} y^{2}}{x y^{3}} \cdot \frac{16 x^{4} y^{2}}{x y^{2}} = 6 x x^{3} \cdot 16 x^{4}.\]

Step 3: Combine Like Terms

Now, let's merge like terms, in this case all \(x\) terms, effectively using the multiplication property of exponents \(a^{m} a^{n} = a^{m+n}\).\[6 x x^{3} \cdot 16 x^{4} = (6\cdot16) x^{(1+3+4)} = 96 x^{8}.\]

Key Concepts

Product-to-Power Rule

The product-to-power rule is an essential tool when dealing with exponents in algebra. It helps simplify expressions that have exponentiation of a product. The rule states that for any numbers or variables, say 'a' and 'b', and a positive exponent 'n', the expression \( (ab)^n \) can be expanded to \((a^n)(b^n)\). This rule is especially handy when you have an expression where a product of two elements is raised to an exponent, as it allows you to separately raise each element to the power and then multiply the result.

For example, in our textbook exercise the term \( (4x^2y)^2 \) is simplified by applying this rule. The base \(4x^2y\) is raised to the power of 2, so we use the product-to-power rule to simplify it into \(16x^4y^2\), where \(4^2 = 16\), \(x^2\) raised to the power of 2 gives us \(x^4\) and \(y^1\) raised to the power of 2 gives us \(y^2\). This expanded form makes it easier to further simplify the expression as we can now treat each variable and constant separately.

Positive Exponents

When working with exponents, it's often required to use only positive exponents in the final expression. A positive exponent tells us how many times to multiply a base number by itself. It's a straightforward concept: for instance, \(x^3\) means \(x \cdot x \cdot x\). A common misunderstanding occurs when dealing with expressions that involve both the numerator and the denominator. A negative exponent indicates that the base is on the wrong side of a fraction. To convert a negative exponent into a positive one, you can flip the base to the other side of the fraction.

In our exercise, we are instructed to use only positive exponents. This emphasizes the importance of rearranging terms such that all exponents are positive in the final answer, as seen with \(96x^8\). Since \(96\) and \(x\) were already positive, and the powers of \(y\) in the numerator and denominator canceled each other out, we reach a solution with positive exponents throughout.

Multiplication Property of Exponents

The multiplication property of exponents is critical in simplifying algebraic expressions that involve multiplying like bases. This property states that when you multiply terms with the same base, you can add the exponents: \(a^m \cdot a^n = a^{m+n}\). This simplification makes it easier to combine terms and reduce the complexity of the expression.

Let's examine this with our sample expression. In the final step of the solution, we combined like terms with the base \(x\). We had \(x\), \(x^3\), and \(x^4\), with exponents 1, 3, and 4, respectively. According to the multiplication property of exponents, we combined them to get \(x^{1+3+4}\), which simplifies to \(x^8\). This, multiplied by the coefficient 96 from combining the constants \(6 \cdot 16\), gave us the final expression \(96x^8\). This illustrates the power of the multiplication property of exponents in streamlining and solving algebraic problems.