Question

Simplify the expression. The simplified expression should have no negative exponents. $$\frac{3 x^{2} y}{2 x} \cdot \frac{6 x y^{2}}{y^{3}}$$

Short answer

The simplified expression becomes \(9x^{2}\)

Step-by-step solution

Consolidate Variables

Rewrite the expression to consolidate the variables. This gives \(\frac{3x^{2}y}{2x} \cdot \frac{6xy^{2}}{y^{3}}\) = \(\frac{3x^{2} \cdot 6x \cdot y \cdot y^{2}}{2x \cdot y^{3}}\)

Simplify Using Power Rules

Simplify the terms by adding/subtracting the exponents of like terms. For the numerator, we have \(x^{2}\) and \(x\), which gives \(x^{2+1}\) or \(x^{3}\). Similarly for \(y\), we have \(y\) and \(y^{2}\), which gives \(y^{1+2}\) or \(y^{3}\). The denominator has an \(x\) and \(y^{3}\), and the constant terms \(3\) and \(6\) divided by \(2\) gives \(9\). This results in \(9x^{3}y^{3} / x \cdot y^{3}\).

Cancel Common Terms

Cancel out the common terms in the numerator and the denominator. This leaves us with the simplified expression \(9x^{2}\)

Key concepts

Negative Exponents

When simplifying algebraic expressions, negative exponents can sometimes cause confusion. A negative exponent represents the reciprocal of the base raised to the opposite positive exponent. In essence, the rule states that any base with a negative exponent can be transformed by taking the reciprocal of the base and then applying the positive exponent.

For example, consider the expression \( x^{-n} \). This can be rewritten as \( \frac{1}{x^n} \). Similarly, \( \frac{y}{x^{-2}} \) is equivalent to \( y \times x^2 \), since moving the base with a negative exponent from the denominator to the numerator changes the sign of the exponent to positive. It's crucial to understand this concept as it ensures the final expression does not contain any negative exponents, as required in many mathematical practices.

Power Rules in Algebra

Dealing with powers in algebra involves a set of rules known as the power rules. These are essential for simplifying expressions, especially when variables are raised to different exponents. One key rule is the product of powers rule which says that when multiplying two powers that have the same base, you can add the exponents. For example, \( x^a \times x^b = x^{a+b} \).

An example from our exercise would be when combining \( x^2 \) and \( x \), the result is \( x^{2+1} \) or \( x^3 \). Another significant rule is the power of a power rule, which states that \( (x^a)^b = x^{a\times b} \). These rules are intuitive once you understand that exponents signify repeated multiplication, and hence, combining powers is a matter of consolidating that repetition.

Consolidating Variables

Consolidating variables is a method used to simplify expressions by combing like terms, which often involves adding or subtracting exponents if the bases are the same. In our exercise, we consolidated the variables by grouping the same bases together. For instance, \( x^2 \) and \( x \) were combined as well as \( y \) and \( y^2 \).

This process makes it easier to apply power rules subsequently. After consolidating, we then need to simplify any coefficients involved, such as multiplying the constants preceding the variables. This is done before moving on to cancelling out any terms that appear in both the numerator and denominator. Consolidating variables not only streamlines the expression but also sets the stage for easier simplification as the next step in the process.