Problem 32 - Algebra 1: Concepts and Skills

Question

Write the quotient in simplest form. $$\frac{16 x^{2}}{8 x} \div \frac{4 x^{2}}{16 x}$$

Step-by-Step Solution

Answer

The simplest form of the given expression is \(8x\).

Step 1: Apply the concept of Keep-Change-Flip

Transform the division of fractions to multiplication by applying the Keep-Change-Flip rule. Therefore, the expression becomes \(\frac{16 x^{2}}{8 x} * \frac{16 x}{4 x^{2}}\).

Step 2: Simplify the Numerators and Denominators

Simplify the fractions before multiplying: \(2x * 4 = 8x\)

Step 3: Apply laws of exponents

When dividing terms with the same base (in this case 'x'), subtract the exponents. \(8x^{1} = 8x\)

Key Concepts

Keep-Change-Flip

When it comes to dividing fractions, the "Keep-Change-Flip" method is your best friend. It transforms division into a more straightforward multiplication problem.

• **Keep** the first fraction as it is.
• **Change** the division sign to a multiplication sign.
• **Flip** the second fraction to its reciprocal.

For the given exercise, you start with \( \frac{16 x^{2}}{8 x} \div \frac{4 x^{2}}{16 x} \). Using Keep-Change-Flip, it becomes \( \frac{16 x^{2}}{8 x} \times \frac{16 x}{4 x^{2}} \). This step is crucial because multiplying fractions is generally easier than dividing, especially when it involves variables like \( x \). By flipping the second fraction, all you are doing is finding its reciprocal, which makes the entire problem simpler to solve.

Laws of Exponents

The laws of exponents play a vital role in simplifying expressions that involve powers of a variable. When you're dealing with exponents, it's important to remember the subtraction rule for division.
For any base \( a \), the rule is: \( a^{m} \div a^{n} = a^{m-n} \). This means when you divide like bases, you subtract the exponents.

• In the given solution, when multiplying \( 16x^2 \cdot \frac{16x}{4x^2} \), you're simplifying by subtracting the exponents of \( x \).
• So, \( x^{2} \div x^{1} = x^{2-1} = x \), which simplifies part of the expression.

Remember, understanding and applying these laws simplifies complex algebraic expressions and is a fundamental skill in algebra.

Fraction Division

Dividing fractions in algebra involves understanding the basic principles of fraction arithmetic combined with algebraic manipulation. The process of dividing one fraction by another can be simplified considerably by following systematic steps.
First, using Keep-Change-Flip, convert the division into multiplication. This makes the problem much easier to handle.
Next, simplify the fractions individually. In our case, \( \frac{16 x^{2}}{8 x} \) simplifies to \( 2x \), and \( \frac{16 x}{4 x^{2}} \) simplifies to \( \frac{4}{x} \).
Finally, multiply these simplified fractions. The product \( 2x \times \frac{4}{x} \) is \( 8 \), after canceling \( x \) from the numerator and the denominator.
Hence, understanding each part of fraction division helps streamline solving algebraic fractions and reduces errors.