Problem 41 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ 2 a^{5} \div \frac{6 a^{2}}{4 b} $$

Step-by-Step Solution

Answer

Question: Perform the following operation and simplify the expression: 2a^5 divided by (6a^2)/(4b). Answer: 4a^3 * (b/3)

Step 1: Simplify the Fractions

First, we will simplify the given expression. The expression is $$ 2 a^{5} \div \frac{6 a^{2}}{4 b}. $$ We can change it into multiplication form by flipping the second fraction (reciprocal) as follows: $$ 2 a^{5} \times \frac{4 b}{6 a^{2}}. $$

Step 2: Simplify the Numerators and Denominators

Now, we simplify the numerator and the denominator by finding the greatest common divisor (GCD) of the numbers and simplifying the variables. $$ = \frac{2}{1} \times \frac{4}{2} a^{5} \times a^{-2} \times \frac{b}{3}. $$

Step 3: Multiply the Fractions

Next, we multiply the fractions component-wise, which means multiplying the numerators together and denominators together: $$ = \frac{2 \times 4}{1 \times 2} a^{5} a^{-2} \frac{b}{3}. $$

Step 4: Final Simplification

Lastly, we simplify the expression by dividing the common terms in the numerator and denominator, and applying the rules for exponents for the variable 'a'. In this case, when multiplying variables with the same base, add the exponents. $$ = \frac{2 \times 4}{1 \times 2} a^{5-2} \frac{b}{3} \Rightarrow 4 a^{3} \frac{b}{3}. $$ So, the final result is: $$ 4 a^{3} \frac{b}{3}. $$

Key Concepts

Multiplication of Fractions

To start with, understanding how to multiply fractions is pivotal for simplifying complex expressions like the one presented. When you multiply fractions, you're essentially finding the product of the numerators and the product of the denominators separately. This is represented as:

• \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)

When multiplying fractions, it's also practical to simplify them beforehand if possible. This involves finding common factors across the numerators and denominators and reducing them. By simplifying early, you can handle smaller numbers, making calculations easier and reducing the likelihood of errors.
In our specific exercise, the fraction \( 2 a^{5} \times \frac{4 b}{6 a^{2}} \) involves multiplying the numerators together and the denominators together. Simply multiply \( 2 \) and \( 4b \) for the numerator, and \( 1 \) and \( 3 \) for the denominator to achieve a simplified result of \( 4 a^{3} \frac{b}{3} \) after further simplification.

Division of Fractions

Fractions can be tricky, but understanding division is made simpler by knowing it can be turned into multiplication. When dividing by a fraction, you multiply by its reciprocal. The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). This means division becomes multiplication as follows:

• \( \frac{p}{q} \div \frac{r}{s} = \frac{p}{q} \times \frac{s}{r} \)

In the original problem, dividing by \( \frac{6 a^{2}}{4 b} \) was converted into multiplying by \( \frac{4 b}{6 a^{2}} \).
This swap to multiplication makes simplifying much easier and aligns with operations you may be more familiar with. Once the conversion is done, follow the steps for multiplying fractions to proceed further. This method maintains accurate results while streamlining calculation steps.

Algebraic Expressions

Handling algebraic expressions involves dealing not only with numbers but also variables like \( a \) and \( b \). Simplifying expressions with variables involves knowing rules for exponents and basic multiplication or division. Exponent rules, particularly when multiplying or dividing like bases, include:

• Multiplying: \( a^m \times a^n = a^{m+n} \)
• Dividing: \( a^m \div a^n = a^{m-n} \)

In our exercise, the expression \( a^{5} \times a^{-2} \) simplifies by subtracting exponents of the base \( a \), resulting in \( a^{5-2} = a^{3} \).
The simplification reflects the same principles applied to any term, numeric or variable-based, ensuring consistency. As variables often come with exponents, being adept at handling them through these rules makes simplification much more straightforward and understandable.