Problem 91 - Algebra for College Students

Question

Perform the indicated operations. $$ \frac{\left(a^{2} b^{3}\right)^{4}}{\left(a b^{4}\right)^{3}} \cdot \frac{(a b)^{3}}{\left(a^{4} b\right)^{2}} $$

Step-by-Step Solution

Answer

The answer is \( b \).

Step 1 - Simplify the First Fraction

Start by simplifying \ \ \(\frac{\left(a^{2} b^{3}\right)^{4}}{\left(a b^{4}\right)^{3}}\).Use the power rule \( (x^m)^n = x^{m \cdot n} \).\[ \frac{\left(a^{2} b^{3}\right)^{4}}{\left(a b^{4}\right)^{3}} = \frac{a^{2 \cdot 4} b^{3 \cdot 4}}{a^{1 \cdot 3} b^{4 \cdot 3}} = \frac{a^8 b^{12}}{a^3 b^{12}}. \]

Step 2 - Cancel Common Terms in the First Fraction

In the fraction, cancel common terms \( b^{12} \) from the numerator and the denominator. Simplify \ \ \( \frac{a^8 b^{12}}{a^3 b^{12}} = \frac{a^8}{a^3} = a^{8-3} = a^5. \)

Step 3 - Simplify the Second Fraction

Now simplify \ \( \frac{(a b)^{3}}{\left(a^{4} b\right)^{2}}. \) Again, use the power rule.\[ \frac{(a b)^{3}}{\left(a^{4} b\right)^{2}} = \frac{a^3 b^3}{a^{4 \cdot 2} b^2} = \frac{a^3 b^3}{a^8 b^2}. \]

Step 4 - Cancel Common Terms in the Second Fraction

In the fraction, cancel common terms \( b^2 \) from the numerator and the denominator. Simplify: \ \( \frac{a^3 b^3}{a^8 b^2} = \frac{a^3}{a^8} \cdot \frac{b^3}{b^2} = a^{3-8} b^{3-2} = a^{-5} b. \)

Step 5 - Multiply the Simplified Fractions

Combine the simplified results from Steps 2 and 4: \( a^5 \cdot a^{-5} b = a^{5-5} b = a^0 b = b. \)

Key Concepts

Exponent Rules

Exponent rules are essential when working with algebraic expressions involving powers. They help us simplify and manipulate expressions efficiently. The most commonly used exponent rules include:

• Product of Powers: \( x^m \times x^n = x^{m+n} \)
• Quotient of Powers: \( \frac{x^m}{x^n} = x^{m-n} \)
• Power of a Power: \( (x^m)^n = x^{m \cdot n} \)
• Power of a Product: \( (xy)^m = x^m y^m \)

These rules help break down complex expressions into simpler forms by systematically applying these principles each step of the way. For example, the power of a power rule simplifies \( \frac{\left( a^2 b^3 \right)^4}{\left( a b^4 \right)^3} \) into \( \frac{ a^8 b^{12} }{ a^3 b^{12} } \), as seen in Step 1.

Simplification

Simplification in algebra often involves reducing expressions to their simplest form. It includes combining like terms, cancelling common factors, and using exponent rules. The primary goal is to make equations and expressions more manageable. In the given example, we use simplification throughout the process:

1. First, we applied the power rule to simplify both the numerator and the denominator.
2. Then, we cancelled out common terms like \( b^{12} \) in the first fraction.
3. Each fraction was reduced further until we reached a simplified form such as \( \frac{a^8}{a^3} = a^5 \). By breaking down each step, we can focus on reducing parts of the expression one at a time, ensuring accuracy and clarity.

Fraction Operations

When dealing with fractions in algebra, it becomes necessary to manipulate both the numerators and denominators appropriately. Fraction operations often involve:

• Finding a common denominator when adding or subtracting fractions.
• Multiplying the numerators and denominators separately when multiplying fractions.
• Dividing by multiplying with the reciprocal of the divisor fraction.

In our example, after simplifying each part of the fraction separately, we combine the results. Specifically, the fractions \( \frac{ \left( a^2 b^3 \right)^4 }{ \left( a b^4 \right)^3 } \) and \( \frac{ \left( a b \right)^3 }{ \left( a^4 b \right)^2 } \) were simplified using exponent rules. Ultimately, multiplying the simplified parts involves straightforward operations: \( a^5 \times a^{-5} b = a^{5-5} b = b \).

Power Rule

The power rule is vital when working with expressions wrapped in parentheses that are raised to an exponent. The power rule states that when a term with an exponent is raised to another power, you multiply the exponents: \( (x^m)^n = x^{m \cdot n} \). The rule simplifies evaluating algebraic expressions significantly.

For instance, given \( \frac{ \left( a^2 b^3 \right)^4 }{ \left( a b^4 \right)^3 } \), we apply the power rule to get \( \frac{ a^{2 \cdot 4} b^{3 \cdot 4} }{ a^{1 \cdot 3} b^{4 \cdot 3} } = \frac{ a^8 b^{12} }{ a^3 b^{12} } \). This simplifies each term correctly. In subsequent steps, the rule was also used to simplify \( \frac{ (a b)^3 }{ \left( a^4 b \right)^2 } = \frac{ a^3 b^3 }{ a^{8} b^2 }\). Applying the power rule correctly leads to easier simplification and manipulation of algebraic expressions.