Problem 66 - Intermediate Algebra

Question

Perform each division. \(\frac{34 s^{30} t^{15}}{14 s^{40} t^{12}}\)

Step-by-Step Solution

Answer

\( \frac{17t^3}{7s^{10}} \)

Step 1: Simplify the Coefficients

The first step is to simplify the coefficients. The coefficient in the numerator is 34 and in the denominator is 14. We divide each by their greatest common divisor which is 2.\[\frac{34}{14} = \frac{34 \div 2}{14 \div 2} = \frac{17}{7}\]. Thus, we have \(\frac{17}{7}\).

Step 2: Simplify the 's' Terms

Next, simplify the terms involving \(s\). We have \(s^{30}\) in the numerator and \(s^{40}\) in the denominator. We can simplify by subtracting the exponents: \(s^{30 - 40} = s^{-10}\).

Step 3: Simplify the 't' Terms

Now simplify the terms involving \(t\). We have \(t^{15}\) in the numerator and \(t^{12}\) in the denominator. Simplifying by subtracting the exponents gives \(t^{15 - 12} = t^3\).

Step 4: Combine the Results

Combine the simplified coefficients and variables into a single expression: \[\frac{17}{7} \times s^{-10} \times t^3 = \frac{17t^3}{7s^{10}}\].

Key Concepts

Polynomial Division

Dividing polynomials involves breaking down expressions that contain terms like variables and coefficients. When dividing expressions like \( \frac{34s^{30}t^{15}}{14s^{40}t^{12}} \), the process starts by simplifying each part of the expression separately.

• Coefficients: These are the numerical parts of the terms. Here, 34 and 14 are the coefficients. Simplifying these by their greatest common divisor (GCD), which is 2, gives us \( \frac{17}{7} \).
• Variable Terms: These include the variables with exponents, like \( s \) and \( t \). To divide these, subtract the exponents of like bases. This is possible because of the rule \( \frac{x^a}{x^b} = x^{a-b} \).

In polynomial division, each component (coefficient and variables) can be simplified one at a time. This makes it easier to handle complex expressions.

Exponents

Exponents are used to express how many times a number called the base is multiplied by itself. They are also vital in polynomial operations. When dividing variables with exponents, it's essential to apply the rule: \( x^a \div x^b = x^{a-b} \).

Here's how it works in our problem:

• For the variable \( s \): We start with \( s^{30} \) in the numerator and \( s^{40} \) in the denominator. Subtract the exponents: \( 30 - 40 = -10 \), so \( s^{-10} \).
• For the variable \( t \): We have \( t^{15} \) in the numerator and \( t^{12} \) in the denominator. Subtracting these gives \( 15 - 12 = 3 \), resulting in \( t^3 \).

Negative exponents can be rewritten using positive exponents. For \( s^{-10} \), it translates into \( \frac{1}{s^{10}} \). This indicates the reciprocal of the base raised to the positive exponent.

Simplifying Fractions

Simplifying fractions is about breaking them down to their simplest form. This includes working with both the coefficients and any present exponents.

• Simplifying Numerical Fractions: We use the greatest common divisor (GCD) to reduce numerical fractions. For example, dividing both 34 and 14 by their GCD (2) results in \( \frac{17}{7} \).
• Simplifying Exponential Fractions: When handling fractions with exponents, subtract the exponents of like bases if present over the numerator and denominator. Apply the rule \( x^a \div x^b = x^{a-b} \).

Finally, we combine these simplified parts into a single expression. With our problem, the result is \( \frac{17 t^3}{7 s^{10}} \). This represents our original fraction in the simplest possible form, easier to interpret and use in further calculations.