Problem 47 - Algebra 2

Question

Simplify each expression. \(\frac{25 a^{2} b^{3}}{6 x^{2} y} \cdot \frac{8 x y^{2}}{20 a^{3} b^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5 b y}{3 a x} \).

Step 1: Write down the given expressions

The expressions to be simplified are \( \frac{25 a^{2} b^{3}}{6 x^{2} y} \) and \( \frac{8 x y^{2}}{20 a^{3} b^{2}} \). Our goal is to multiply these two fractions and simplify the resulting expression.

Step 2: Multiply the numerators and denominators

To multiply fractions, we multiply the numerators and the denominators: \( (25 a^{2} b^{3})(8 x y^{2}) \) for the numerator and \( (6 x^{2} y)(20 a^{3} b^{2}) \) for the denominator. This gives: \[\frac{25 \cdot 8 \cdot a^{2} \cdot b^{3} \cdot x \cdot y^{2}}{6 \cdot 20 \cdot x^{2} \cdot y \cdot a^{3} \cdot b^{2}}.\]

Step 3: Simplify by cancelling common factors

First, multiply the numerical parts: \(25 \cdot 8 = 200\) and \(6 \cdot 20 = 120\). Factoring these numbers, we have \(200 = 2^{3} \cdot 5^{2}\) and \(120 = 2^{3} \cdot 3 \cdot 5\). Cross out the common factors: \(2^{3}\) and \(5\). Then simplify the variable parts:* For \(a\): \(a^{2}\) in the numerator and \(a^{3}\) in the denominator; cancel to leave \(a\) in the denominator.* For \(b\): \(b^{3}\) in the numerator and \(b^{2}\) in the denominator; cancel to leave \(b\) in the numerator.* For \(x\): \(x\) in the numerator and \(x^{2}\) in the denominator; cancel to leave \(x\) in the denominator.* For \(y\): \(y^{2}\) in the numerator and \(y\) in the denominator; cancel to leave \(y\) in the numerator.Thus, the expression simplifies to:\[\frac{5 \cdot b \cdot y}{3 \cdot a \cdot x} = \frac{5 b y}{3 a x}.\]

Step 4: Verify simplification

Ensure no further simplification is possible by checking that all common factors are canceled, which confirms the expression \( \frac{5 b y}{3 a x} \) is in its simplest form.

Key Concepts

Fraction Multiplication

When dealing with fraction multiplication, the process begins by multiplying both numerators to form the new numerator and also doing the same with the denominators to form the new denominator. This is straightforward because, unlike addition or subtraction of fractions, there is no need to find a common denominator beforehand. Instead, we directly compute the product:

• The numerator is the product of the numerators of the involved fractions.
• The denominator is the product of the denominators of the involved fractions.

For example, if you have two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), their product will be \(\frac{a \cdot c}{b \cdot d}\). In the given exercise, \[\frac{25 a^{2} b^{3}}{6 x^{2} y} \cdot \frac{8 x y^{2}}{20 a^{3} b^{2}}\] we accomplish multiplication by placing the products in the numerator and the denominator, resulting in a new single fraction.

Simplifying Expressions

Simplifying algebraic expressions, especially those involving fractions, involves reducing them to their simplest form. This is done by identifying and canceling shared factors from the numerator and the denominator wherever possible. To approach simplification:

• First, simplify any numeric coefficients by finding their greatest common divisor (GCD).
• Next, inspect the powers of each variable in the numerator and the denominator, reducing them where applicable.

In our example, we calculate the product of the coefficients, literally writing them as their prime factorizations:- \(200 = 2^{3} \cdot 5^{2}\)- \(120 = 2^{3} \cdot 3 \cdot 5\)Here, the common factors \(2^3\) and \(5\) cancel out. After dealing with coefficients, apply the same concept to each variable, reducing power by subtracting powers if they appear in both the numerator and the denominator.

Common Factor Cancellation

Common factor cancellation is a critical step in simplifying expressions. It involves recognizing and removing identical factors from the numerator and denominator, based on their presence in both, essentially "canceling" them out. This simplifies the fraction without changing its value.

• For common numerical factors, factorize both the numerator and denominator into primes and then cancel common terms.
• With variables, subtract the smaller exponent from the larger one to find out how much of a variable remains in either the numerator or denominator.

Let's illustrate this with the expression:- For the factor \(a\), we have \(a^2\) in the numerator and \(a^3\) in the denominator, leaving \(a\) as \(a^{3-2} = a^1\) in the denominator.- With \(b\), \(b^3\) - \(b^2\) leaves \(b\) in the numerator.- Similarly, \(x\) and \(y\) result in remaining \(x\) in the denominator and \(y\) in the numerator, respectively.Ultimately, simplifying ensures the expression is as compact as possible, represented by: \[\frac{5 b y}{3 a x}\] which includes only the factors that couldn't be canceled away.