Problem 73 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\( \frac{a^{6} b^{2}}{-8 c^{2}} \)

Step 1 - Simplify the numerator and denominator

First, simplify the expressions inside the parentheses and apply the power to each term: \[\frac{\left(a^{2} b^{3} c\right)^{2}}{\left(-2 a b^{2} c\right)^{3}} \cdot \frac{\left(a^{3} b^{2} c\right)^{3}}{(a b c)^{4}}\]This gives:\[\frac{a^{4} b^{6} c^{2}}{-8 a^{3} b^{6} c^{3}} \cdot \frac{a^{9} b^{6} c^{3}}{a^{4} b^{4} c^{4}} \]

Step 2 - Simplify each fraction

Now, simplify each fraction by canceling out common factors in the numerator and denominator: For the first fraction:\[\frac{a^{4} b^{6} c^{2}}{-8 a^{3} b^{6} c^{3}} = \frac{a^{4-3} b^{6-6} c^{2-3}}{-8} = \frac{a}{-8 c} \text{ (since } b^{6-6} = 1) \]For the second fraction:\[\frac{a^{9} b^{6} c^{3}}{a^{4} b^{4} c^{4}} = \frac{a^{9-4} b^{6-4} c^{3-4}} = a^{5} b^{2} c^{-1} = \frac{a^{5} b^{2}}{c}\]

Step 3 - Combine the simplified fractions

Now, combine the simplified fractions: \[\frac{a}{-8 c} \cdot \frac{a^{5} b^{2}}{c} = \frac{a \cdot a^{5} b^{2}}{-8 c \cdot c} = \frac{a^{6} b^{2}}{-8 c^{2}}\]

Step 4 - Final Simplification

Consolidate the terms to get the final simplified form of the expression: \[\frac{a^{6} b^{2}}{-8 c^{2}}\]This is the simplest form of the given expression.

Key Concepts

Exponent Rules

Understanding and applying exponent rules is crucial in simplifying algebraic expressions. When you raise a power to another power, you multiply the exponents. For example, \((a^3)^2 = a^{3 \times 2} = a^6\). Another important rule is the product of powers where you add the exponents, like \(a^2 \times a^3 = a^{2+3} = a^5\). Lastly, when dividing powers with the same base, subtract the exponents: \((a^5)/(a^2) = a^{5-2} = a^3\). In the given problem, these rules help transform complex terms into simpler forms.

Fraction Simplification

To simplify fractions in algebra, break down each term in the numerator and the denominator. Extract common factors and cancel them out. For instance, \((a^4 b^6 c^2)/(-8 a^3 b^6 c^3)\) simplifies by subtracting the exponents of like bases. This gives: \(a^{4-3} b^{6-6} c^{2-3} / -8 = a / (-8c)\), as \(b^{6-6} = 1\). To make this easier:

• Always factor out common terms.
• Simplify step-by-step to prevent mistakes.
• Use exponent rules to combine or divide terms.

These principles also apply to the second fraction in the exercise, leading to the simplification: \(a^5 b^2 / c\).

Algebraic Operations

Performing algebraic operations involves combining like terms and applying multiplication or division rules correctly. After simplifying the fractions, you'll need to multiply them together. For instance, \( a / (-8c) \times (a^5 b^2 / c) \). Combine the numerators and denominators separately:
\( a \times a^5 b^2 = a^6 b^2 \) (because \( a \times a^5 = a^6\))
\( -8c \times c = -8c^2 \) (since \( c \times c = c^2 \)).
This results in: \((a^6 b^2) / (-8c^2)\). Following these steps ensures a clear path from a complex algebraic expression to its simplest form.