Problem 75 - Algebra for College Students

Question

Perform the indicated operations. $$\frac{(2 m n)^{3}}{6 m n^{2}} \div \frac{2 m^{2} n^{3}}{\left(m^{2} n\right)^{4}}$$

Step-by-Step Solution

Answer

\frac{2m^8n^2}{3}.\

Step 1 - Simplify the Numerator of the First Fraction

The numerator of the first fraction is \((2mn)^3\). Expand this as follows: \((2mn)^3 = 2^3 \times m^3 \times n^3 = 8m^3n^3\).

Step 2 - Simplify the First Fraction

Divide the simplified numerator by the denominator \(6mn^2\): \frac{8m^3n^3}{6mn^2} = \frac{8}{6} \times \frac{m^3}{m} \times \frac{n^3}{n^2} = \frac{4}{3}m^2n.\

Step 3 - Simplify the Denominator of the Second Fraction

The denominator of the second fraction is \((m^2n)^4\). Expand this as follows: \((m^2n)^4 = m^{8} n^4\).

Step 4 - Simplify the Second Fraction

Divide the numerator by the denominator \(m^8 n^4\): \frac{2m^2n^3}{m^8 n^4} = \frac{2}{1} \times \frac{m^2}{m^8} \times \frac{n^3}{n^4} = 2m^{-6}n^{-1} = \frac{2}{m^6n}.\

Step 5 - Divide the Simplified First Fraction by the Simplified Second Fraction

Perform the division \(\frac{4}{3}m^2n \div \frac{2}{m^6n}\): Write this as a multiplication by the reciprocal: \frac{4}{3}m^2n \times \frac{m^6n}{2} = \frac{4 \times m^2 \times m^6 \times n \times n}{3 \times 2} = \frac{4m^8n^2}{6} = \frac{2m^8n^2}{3}. \

Key Concepts

Fraction Simplification

Simplifying fractions means making the fraction as simple as possible. This often involves canceling common factors in the numerator (top number) and the denominator (bottom number). You look for any common factors and divide both the numerator and the denominator by those factors.
For example, in the exercise given: \[ \frac{8m^3n^3}{6mn^2} = \frac{8}{6} \times \frac{m^3}{m} \times \frac{n^3}{n^2} \] Here's how to simplify it:

• First, recognize that 8 and 6 have a common factor of 2. So, divide both by 2 to get \( \frac{4}{3} \).
• Next, simplify \( \frac{m^3}{m} = m^{3-1} = m^2 \).
• Finally, simplify \( \frac{n^3}{n^2} = n^{3-2} = n \).

This gives you the simplified fraction: \( \frac{4}{3}m^2 n \) .

Exponent Rules

Understanding exponent rules is crucial for simplifying expressions that involve powers. Some key rules include:

• Product of powers: When multiplying like bases, add the exponents. For example, \( x^a \cdot x^b = x^{a+b} \).
• Power of a power: When raising a power to another power, multiply the exponents. For example, \( (x^a)^b = x^{a \cdot b} \).
• Power of a product: Distribute the exponent to both the base and the exponent. For example, \( (xy)^a = x^a \cdot y^a \).
• Negative exponents: A negative exponent means the reciprocal of the base raised to the opposite positive exponent. For example, \( x^{-a} = \frac{1}{x^a} \).

In the exercise, you use these rules extensively:

• For \( (2mn)^3 \), distribute the exponent: \( 2^3 \cdot m^3 \cdot n^3 = 8m^3n^3 \)
• For \( (m^2n)^4 \), distribute the exponent: \( (m^2)^4 \cdot n^4 = m^{2 \cdot 4} \cdot n^4 = m^8n^4 \)

Division of Fractions

Dividing fractions involves multiplying by the reciprocal of the divisor. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
For example, to divide \( \frac{a}{b} \div \frac{c}{d} \), you multiply by the reciprocal: \( \frac{a}{b} \times \frac{d}{c} \).
In the given exercise:
\[ \frac{4}{3}m^2 n \div \frac{2}{m^6n} \]
You can rewrite this as:
\[ \frac{4}{3} m^2 n \times \frac{m^6 n}{2} \]
Next, perform multiplication:

• Combine constants: \( \frac{4 \times m^6 n \times m^2 n}{3 \times 2} \)
• Combine like terms: \( \frac{4 m^8 n^2}{6} \)
• Simplify constant factor: \( \frac{4}{6} = \frac{2}{3} \)

This results in: \( \frac{2 m^8 n^2}{3} \).

Numerator and Denominator

The numerator is the top number of a fraction, while the denominator is the bottom number. They show how many parts of a whole you have and into how many parts the whole is divided, respectively.
Consider the fractions involved in the exercise:
In the fraction \( \frac{8m^3n^3}{6mn^2} \):

• The numerator is \( 8m^3n^3 \), meaning eight parts of an expression involving three multiplications of m and n each.
• The denominator is \( 6mn^2 \), meaning six parts involving one multiplication of m and two multiplications of n.

The same logic applies to other fractions in the exercise. Understanding the roles of numerator and denominator helps in operations and simplifying fractions accurately.