Problem 58 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ 9 k \div \frac{27 k^{4}}{4} $$

Step-by-Step Solution

Answer

\[ \frac{4}{3k^3} \]

Step 1 - Rewrite the Division

Rewrite the division as a multiplication problem by taking the reciprocal of the divisor. The problem is initially given as \[ 9k \div \frac{27k^4}{4} \] which can be rewritten as \[ 9k \times \frac{4}{27k^4} \]

Step 2 - Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \[ \frac{ 9k \times 4 }{ 1 \times 27k^4 } = \frac{ 36k }{ 27k^4 } \]

Step 3 - Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 36 and 27 is 9: \[ \frac{36k \div 9}{27k^4 \div 9} = \frac{4k}{3k^4} \]

Step 4 - Simplify the Variable Part

Simplify the variable part by subtracting the exponent of the denominator from the exponent of the numerator. Since \(k\) in the numerator has an exponent of 1 and the denominator has an exponent of 4, it simplifies as follows: \[ \frac{4k}{3k^4} = \frac{4}{3k^{4-1}} = \frac{4}{3k^3} \]

Key Concepts

Division of Fractions

When you're given a problem that involves the division of fractions, it's often easier to convert it into a multiplication problem. This is done by taking the reciprocal of the divisor (the fraction you are dividing by) and then multiplying. For example, if you have \(\frac{a}{b} \div \frac{c}{d}\), this can be rewritten as \(\frac{a}{b} \times \frac{d}{c} \). This method simplifies calculations and helps avoid mistakes.

Reciprocal

A reciprocal is simply flipping a fraction. That means you switch the numerator (the top number) and the denominator (the bottom number). For instance, the reciprocal of \(\frac{3}{4} \) is \(\frac{4}{3} \). This concept is particularly useful when dividing fractions because it allows you to turn the division operation into multiplication, which can be much easier to handle.

Simplifying Exponents

Simplifying exponents is another important step in algebraic simplification. When you have the same base with different exponents in the numerator and the denominator, you can subtract the exponent of the denominator from the exponent of the numerator. For example, simplifying \(k^a \div k^b\) gives you \(k^{a-b} \). This rule helps us to keep expressions with exponents more manageable.

Greatest Common Divisor

The greatest common divisor (GCD) helps to simplify fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. For example, the GCD of 36 and 27 is 9 because 9 is the largest number that divides both of them exactly. By dividing both the numerator and the denominator of a fraction by their GCD, you simplify the fraction to its simplest form.