Problem 96 - Elementary Algebra

Question

\(\frac{110 x^{9}}{300 x^{5}}\)

Step-by-Step Solution

Answer

\[ \frac{11 x^{4}}{30} \]

Step 1 - Simplify the Coefficients

Divide the coefficients of the numerator and the denominator. Here, divide 110 by 300. \[ \frac{110}{300} = \frac{11}{30} \]

Step 2 - Simplify the Variables

Subtract the exponent of the variable in the denominator from the exponent of the variable in the numerator. \[ x^{9 - 5} = x^{4} \]

Step 3 - Combine Results

Combine the simplified coefficients and variables. \[ \frac{11}{30} \times x^{4} = \frac{11 x^{4}}{30} \]

Key Concepts

simplify coefficients

To start simplifying rational expressions, focus on the coefficients - the numerical parts of the expression. In our exercise, we have the fraction \(\frac{110}{300}\).
One of the first steps is to divide these numbers.
Simplifying coefficients involves breaking down numbers into their simplest forms by finding their greatest common divisor (GCD).
For our problem: \[ \frac{110}{300} \]
We can see that both 110 and 300 are divisible by their GCD, which is 10.
Dividing both the numerator and the denominator by 10, we get:
\[ \frac{110 \div 10}{300 \div 10}= \frac{11}{30} \]
Now the coefficients are in their simplest form: \(\frac{11}{30}\)

• Always divide by the GCD for simplicity.
• If stuck, break down numbers into prime factors to find the GCD.

subtract exponents

Next, we simplify the variables by subtracting their exponents.
In the given exercise, we have variables with exponents in both the numerator and the denominator: \[ \frac{x^{9}}{x^{5}} \]
To simplify, subtract the exponent in the denominator from the exponent in the numerator: \[ x^{9 - 5} = x^{4} \]
This operation works because of the rules of exponents: when dividing like bases, subtract the exponents.
Remember these key points:

• Ensure the bases are the same before subtracting exponents.
• Negative exponents indicate reciprocal. For instance, \(x^{-n}= \frac{1}{x^{n}}\).

combine results

Finally, after simplifying both the coefficients and the exponents, it's time to combine them.
In our exercise, we have \(\frac{11}{30}\) as the simplified coefficient and \(x^{4}\) as the simplified variable.
To combine these:
Multiply the simplified coefficient by the simplified variable:
\[ \frac{11}{30} \times x^{4} = \frac{11 x^{4}}{30} \]
Therefore, the fully simplified expression is \(\frac{11 x^{4}}{30}\).

• Always ensure that coefficients and variables are simplified separately first.
• Combining results should be straightforward multiplication.