Question

Find the product. $$ \frac{48 x^5 y^3}{y^4} \cdot \frac{x^2 y}{6 x^3 y^2} $$

Short answer

The solution is \( \frac{8x^4}{y} \)

Step-by-step solution

Cancel Terms

First, let's cancel out similar terms from the numerator and denominator where we can. The fraction to cancel out from the first term is \(y^3/y^4 = 1/y \) and for the second term is \(x^3 x^2 = x^{-1}\) and \(y / y^2 = y^{-1}\). The new expression then becomes: \( \frac{48x^5}{y} \cdot \frac{x^{-1} y^{-1}}{6} \) .

Multiply Fractions

Next, after cancelling the same terms in both fractions, they can be multiplied. When multiplying fractions, the numerators are multiplied with each other and the denominators are multiplied with each other. So, \( \frac{48x^5}{y} \times \frac{x^{-1} y^{-1}}{6} = \frac{48x^5 \cdot x^{-1}}{y \cdot 6 \cdot y^{-1}} \).

Simplify

After multiply these, it leaves us with the expression \( \frac{48x^5 \cdot x^{-1}}{6y} \) It simplifies to \( \frac{48x^4}{6y} \) . Then we divide 48 by 6, which leaves us with \( \frac{8x^4}{y} \)

Key concepts

Algebraic Expression Simplification

Simplifying algebraic expressions is crucial for making complex problems easier to handle. An algebraic expression could include variables, numbers, and operation symbols. When simplifying, we aim to combine like terms, use the distributive property to eliminate parentheses, and reduce fractions to their simplest form.

For example, consider the expression \(48x^5 y^3/y^4 \cdot x^2 y/6x^3 y^2\). To simplify, factors present in both the numerator and the denominator can be divided out. Moreover, when simplifying fractions, we seek to reduce numerical coefficients — in this case, dividing 48 by 6 to simplify the fraction to an equivalent form with smaller numbers, making it \(8x^5 y^3/y^4 \cdot x^2 y/6x^3 y^2\). Through these steps, we convert a complex fraction into a less intimidating form that's easier to interpret and solve.

Canceling Terms in Fractions

Canceling terms in fractions is a method used to simplify fractions by reducing them to their lowest terms. In algebraic fractions, canceling involves dividing both the numerator and the denominator by the same algebraic terms or factors.

In the given exercise, we see that variables in the numerator and denominator can be canceled out. For instance, \(y^3/y^4 \rightarrow 1/y\) and \(x^3 \cdot x^2 = x^{3+2} = x^5\) which simplifies further when combined with \(x^5\) in the opposite fraction to \(x^5 \cdot x^{-1} = x^{5-1} = x^4\). This process is essential because it helps reduce the expression to its simplest form, making the multiplication easier to perform and the solution easier to find.

Exponent Rules

Understanding exponent rules is essential when working with algebraic expressions that have powers or exponents. Some key rules include the product of powers rule, which states that when multiplying like bases, we add their exponents (\(x^a \cdot x^b = x^{a+b}\)). Similarly, when dividing like bases, we subtract the exponents (\(x^a / x^b = x^{a-b}\)).

In our problem, applying the exponent rules simplifies the multiplication of terms with exponents. As we multiply \(48x^5 \cdot x^{-1}\), we use the product of powers rule, combining the exponents: \(x^{5+(-1)} = x^4\). Correctly applying these rules results in a simplified expression that is \(8x^4/y\). This step is critical for achieving the correct answer efficiently and accurately.