Problem 72 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{-4 a^{3}}{3 b} \div \frac{2 a}{6 b^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\frac{-4 a^{3}}{3 b} \div \frac{2 a}{6 b^{2}}\). Answer: \(6 a^{3} b\)

Step 1: Rewrite the division as multiplication

To solve this problem, we'll first rewrite the division as multiplication by finding the reciprocal of the second fraction, then multiply the first fraction by the reciprocal of the second fraction. Remember that the reciprocal of a fraction is obtained by swapping its numerator and denominator. So, the reciprocal of \(\frac{2a}{6b^2}\) is \(\frac{6b^2}{2a}\). $$\frac{-4 a^{3}}{3 b} \div \frac{2 a}{6 b^{2}} = \frac{-4 a^{3}}{3 b} \cdot \frac{6 b^2}{2 a}$$

Step 2: Simplify and cancel terms

Now, we will simplify and cancel any common terms in the numerators and denominators: $$\frac{-4 a^{3}}{3 b} \cdot \frac{6 b^2}{2 a} = \frac{(-4\cancelto{2}{a^3})(3\cancel{b}^{2})}{\cancel{3} b(\cancel{2}\cancel{a})}$$

Step 3: Perform multiplication

Now we can multiply the simplified numerators and denominators: $$\frac{2 a^{3} \cdot 3 b^{2}}{b}$$

Step 4: Simplify the result

Now, simplify the expression by canceling common term \(b\) from the numerator and the denominator: $$\frac{2 a^{3} \cdot 3 \cancel{b}^{2}}{\cancel{b}} = 6 a^{3} b$$ The simplified result of the given expression is \(6 a^{3} b\).

Key Concepts

Fraction Division

When working with fraction division, it's important to understand that dividing by a fraction is equivalent to multiplying by its reciprocal. This concept might sound unfamiliar at first, but it makes sense when we break it down:

• Take the reciprocal of the second fraction: Flip the numerator and the denominator.
• Convert the division operation into a multiplication operation.

Let's consider the initial exercise:\[\frac{-4 a^{3}}{3 b} \div \frac{2 a}{6 b^{2}}\]By following the steps:

• Flip the fraction to get its reciprocal: The reciprocal of \(\frac{2a}{6b^2}\) becomes \(\frac{6b^2}{2a}\).
• Switch the division to multiplication: \(\frac{-4 a^{3}}{3 b} \times \frac{6 b^2}{2 a}\)

This transformation allows us to proceed with multiplying the fractions instead of dividing.

Simplifying Fractions

Simplifying fractions is a crucial step to make mathematical expressions more manageable and easier to interpret. In essence, simplification involves canceling out any common factors present in both the numerator and the denominator.

• Identify common factors and cancel them if possible.
• Remember that canceling is essentially dividing both the numerator and the denominator by the same number.

Taking the expression:\[\frac{-4 a^{3}}{3 b} \times \frac{6 b^2}{2 a}\]To simplify, follow these steps:

• Cancel common numerical factors, like \(2\) and \(3\), and coefficients like \(a\). In this case:
• The term \(-4\) divided by \(2\) leaves \(-2\).
• Cancel one \(a\) from both \(a^3\) and \(a\), making it \(a^2\).
• Cancel \(b\) from both \(b^2\) and \(b\), leaving \(b\).

The result of these operations is a much more streamlined expression. Ultimately, simplifying helps achieve the clean solution:\[6 a^{3} b\]

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is important, not just for division problems but for many algebraic manipulations. To find a reciprocal:

• Swap the numerator and the denominator of the fraction.
• The process is theoretical but incredibly useful in solving equations.

In the exercise, the fraction \(\frac{2a}{6b^2}\) has a reciprocal of \(\frac{6b^2}{2a}\).

• By flipping the fraction, you essentially set up new opportunities to multiply and simplify.
• This concept plays a critical role in changing division problems into multiplication ones, which are generally more straightforward to solve.

Reciprocals turn potential division difficulties into multiplication steps, easing the path to solutions.