Problem 35 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(\frac{8 a^{3} b^{2}}{16 a^{2} b^{3}}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{a^4}{16b^4} \).

Step 1: Simplify the Fraction

First, simplify the fraction inside the parentheses. We have \( \frac{8 a^3 b^2}{16 a^2 b^3} \). Simplify by dividing both the numerator and the denominator by their greatest common divisor.- Divide 8 by 16: \( \frac{8}{16} = \frac{1}{2} \).- Divide the variables: \( a^3 \div a^2 = a \) and \( b^2 \div b^3 = \frac{1}{b} \).Thus, the simplified expression is \( \frac{a}{2b} \).

Step 2: Apply the Power to the Fraction

Now, apply the power of 4 to the simplified fraction \( \left( \frac{a}{2b} \right)^4 \). This means we raise both the numerator and the denominator to the power of 4:\[ \left( \frac{a}{2b} \right)^4 = \frac{a^4}{(2b)^4} = \frac{a^4}{2^4 b^4} \].

Step 3: Simplify the Exponentiation

Calculate \( 2^4 \) since the denominator contains \( (2b)^4 \).\[ 2^4 = 16 \].Thus, the fraction becomes \( \frac{a^4}{16 b^4} \).

Step 4: Final Simplified Expression

Check the simplified expression \( \frac{a^4}{16 b^4} \) to ensure all steps were applied correctly. The fraction is already in its simplest form, as there are no common factors between the numerator and the denominator.

Key Concepts

Fraction Simplification

Fraction simplification is a key concept in algebra that involves reducing a fraction to its simplest form. In the given exercise, the expression \( \frac{8 a^3 b^2}{16 a^2 b^3} \) needs to be simplified. The first step is to divide both the numerator and the denominator by their greatest common divisor. This means finding numbers and variables that can evenly divide both parts of the fraction.

• Divide 8 by 16, which simplifies to \( \frac{1}{2} \).
• For the variables, divide \( a^3 \) by \( a^2 \), ending with \( a \), and \( b^2 \) by \( b^3 \), resulting in \( \frac{1}{b} \).

By doing so, the fraction simplifies to \( \frac{a}{2b} \). Simplification helps us work with expressions in a more manageable form and is crucial for solving algebraic problems efficiently.

Exponentiation

Exponentiation is a powerful mathematical operation that involves raising a number or an expression to a power. In the exercise, once the fraction \( \frac{a}{2b} \) was simplified, it was raised to the fourth power: \( \left( \frac{a}{2b} \right)^4 \).

• To apply the power, raise both the numerator and the denominator individually to the power of 4.
• The numerator becomes \( a^4 \).
• The denominator becomes \( (2b)^4 \), which can be calculated as \( 2^4 b^4 \).

Finally, calculate \( 2^4 \), which equals 16, resulting in a denominator of \( 16b^4 \). Exponentiation is essential for expanding expressions and is widely used in various fields, including physics and engineering.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. In simplifying algebraic fractions, understanding the properties of variables and coefficients is critical. The given problem involves an algebraic fraction, \( \frac{8 a^3 b^2}{16 a^2 b^3} \), where the goal was to reduce it to its simplest form.Key steps include:

• Identifying and canceling common factors.
• Applying the laws of exponents during simplification and exponentiation.
• Ensuring the final expression has no common factors remaining to achieve the simplest form, as seen in \( \frac{a^4}{16 b^4} \).

Understating algebraic fractions allows students to handle complex rational expressions more easily and is foundational for higher-level algebra concepts.