Problem 8 - Algebra 2

Question

Multiply. State any restrictions on the variables. $$ \frac{2 x^{4}}{10 y^{2}} \cdot \frac{5 y^{3}}{4 x^{3}} $$

Step-by-Step Solution

Answer

The answer is \(x/4y\), where y cannot be zero.

Step 1: Standardize fractions

First, remember that when you multiply fractions, you just multiply the numerators together to get your final numerator, and multiply the denominators together to get the final denominator. Hence, you write the expression as \((2x^{4})(5y^{3})/(10y^{2} * 4x^3)\)

Step 2: Simplify the fractions

Perform the multiplication in both the numerator and the denominator and you will get \(10x^{4}y^{3}/40x^{3}y^{2}\). After this you can simplify it by reducing the variables and numbers. This gives \(x/4y\).

Step 3: Restrictions on variables

Finally, after the fraction is fully simplified, you state the restrictions on the variables. Here, the variable y cannot be equal to 0 because it is located in the denominator of the fraction. If it were equal to 0, the expression would be undefined.

Key Concepts

Variable Restrictions

When working with algebraic fractions, it's crucial to identify any variable restrictions. This helps in avoiding mathematical errors, like dividing by zero, which makes an expression undefined. Consider an expression like \( \frac{1}{x} \). Here, \( x \) cannot be zero. If it were zero, dividing by zero would occur, which isn't allowed in mathematics.
In the original exercise, the fraction \( \frac{2x^4}{10y^2} \cdot \frac{5y^3}{4x^3} \) introduces the variable \( y \). Since \( y \) is in the denominator, it cannot be zero. Hence, the restriction on \( y \) is that it must not equal zero. Checking for these restrictions before proceeding with simplification or multiplication ensures the integrity of the solution.

Simplifying Fractions

Once you have set any restrictions, the next step is simplifying the algebraic fractions. Simplification involves reducing the fraction to its simplest form by canceling common factors in the numerator and the denominator.
For instance, if you have a fraction like \( \frac{6x^3}{9x^2} \), you can divide both the numerator and the denominator by their greatest common factor. Here, that factor is \( 3x^2 \). Dividing top and bottom by \( 3x^2 \) gives \( \frac{2x}{3} \).
In the original exercise, after multiplying the numerators and denominators, the expression \( \frac{10x^4y^3}{40x^3y^2} \) can be simplified. Both 10 and 40 are divisible by 10, and all terms have common \( x \) and \( y \) factors. Simplifying leads to \( \frac{x}{4y} \), which is the simplest form of the algebraic fraction.

Multiplying Numerators and Denominators

Multiplying algebraic fractions involves straightforward operations on numerators and denominators. Like multiplying two normal fractions, you multiply the numerators together and then the denominators.

• Write out the expression, ensuring all components are clearly marked. For example, \( (a/b) \cdot (c/d) \).
• Multiply the numerators: \( a \cdot c \).
• Multiply the denominators: \( b \cdot d \).
• Combine these into a new fraction: \( \frac{ac}{bd} \).

In the original problem, multiplying the numerators \( 2x^4 \) and \( 5y^3 \) gives \( 10x^4y^3 \). Then, multiplying the denominators \( 10y^2 \) and \( 4x^3 \) results in \( 40x^3y^2 \). This operation sets the stage for further simplification of the fraction.