Problem 44 - Elementary and Intermediate Algebra

Question

Divide, and then simplify, if possible. \(\frac{21 x}{z^{2}} \div \frac{7 x^{3}}{z^{5}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{3z^3}{x^2}\).

Step 1: Rewrite the Division as Multiplication

To simplify division involving fractions, we convert it into multiplication by taking the reciprocal of the divisor. The given expression is \(\frac{21x}{z^{2}} \div \frac{7x^3}{z^5}\). When changing to multiplication, this becomes \(\frac{21x}{z^{2}} \times \frac{z^5}{7x^3}\).

Step 2: Multiply the Fractions

Now, multiply the numerators together and the denominators together. The expression is \(\frac{21x \cdot z^5}{z^2 \cdot 7x^3}\).

Step 3: Simplify the Expression

We simplify by cancelling out the common terms between the numerator and the denominator. Simplifying \(\frac{21x \cdot z^5}{z^2 \cdot 7x^3}\):- The 7 in the denominator cancels with the 21 in the numerator, leaving 3.- The \(x\) terms simplify as \(x\) in the numerator and \(x^3\) in the denominator result in \(x^{3-1} = x^2\) in the denominator.- The \(z\) terms simplify as \(z^5\) in the numerator and \(z^2\) in the denominator result in \(z^{5-2} = z^3\) in the numerator.The resulting expression is \(\frac{3z^3}{x^2}\).

Key Concepts

Fraction Division

Dividing fractions can be confusing, but a simple rule makes it much easier. When you divide fractions, instead of dividing, you multiply by the reciprocal of the second fraction. Here's how it works: Start with your division problem. For example, you're asked to divide \(\frac{21x}{z^2}\) by \(\frac{7x^3}{z^5}\). Instead of dividing directly, flip the second fraction and multiply.

• The original division expression: \(\frac{21x}{z^2} \div \frac{7x^3}{z^5}\)
• Becomes a multiplication expression: \(\frac{21x}{z^2} \times \frac{z^5}{7x^3}\)

Now, you just need to multiply across the numerators and denominators. It's a little trick that can really simplify the process of working with fractions in algebraic expressions.

Simplification

Simplifying fractions means reducing them to their simplest form. This involves canceling out common terms between the numerator and the denominator until no further reduction is possible. Let's look at what happens when you simplify the fraction you get after multiplying:

• Your expression after multiplication: \(\frac{21x \cdot z^5}{z^2 \cdot 7x^3}\)
• First, simplify the numbers: \(21\) divided by \(7\) leaves \(3\).

Next, move to the variables. You can simplify \(x\) and \(z\) terms using the properties of exponents:

• The \(x\) terms: \(x^{1}\) in the numerator and \(x^{3}\) in the denominator simplify to \(x^{3-1} = x^{2}\) in the denominator.
• The \(z\) terms: \(z^{5}\) in the numerator and \(z^{2}\) in the denominator simplify to \(z^{5-2} = z^{3}\) in the numerator.

After all simplifications, your fraction becomes \(\frac{3z^3}{x^2}\). It is crucial to ensure you simplify fully to arrive at the neatest form of the fraction.

Reciprocals

The concept of reciprocals is fundamental in dividing fractions and is a game changer in simplifying problems. A reciprocal is simply what you multiply a number by, to get one. For any fraction, its reciprocal is obtained by swapping its numerator and denominator. So, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).Applying this to the given exercise:

• The fraction \(\frac{7x^3}{z^5}\) has a reciprocal of \(\frac{z^5}{7x^3}\).
• When you divide by \(\frac{7x^3}{z^5}\), you actually multiply by \(\frac{z^5}{7x^3}\).

Understanding reciprocals allows you to switch between division and multiplication effortlessly. It's a powerful technique that transforms complex fraction problems into more manageable multiplication problems.