Problem 98 - Elementary Algebra

Question

\(\frac{5 p^{8}}{100 p^{13}}\)

Step-by-Step Solution

Answer

\frac{1}{20p^5}

Step 1 - Simplify the Coefficients

First, separate the numerical coefficients from the variable terms. The fraction for the numerical coefficients is \ \( \frac{5}{100} \ \).

Step 2 - Reduce the Numerical Coefficient

Simplify \ \( \frac{5}{100} \ \) by dividing the numerator and the denominator by their greatest common divisor, which is 5. This gives \ \( \frac{5 \div 5}{100 \div 5} = \frac{1}{20} \ \).

Step 3 - Simplify the Variable's Exponents

For the variable terms \ \( \frac{p^8}{p^{13}} \ \), use the property of exponents \ \( \frac{a^m}{a^n} = a^{m-n} \ \) to get \ \( p^{8-13} = p^{-5} \ \).

Step 4 - Combine Results

Combine the simplified coefficients and variable terms: \ \( \frac{1}{20} \times p^{-5} = \frac{1}{20p^5} \ \).

Key Concepts

fraction simplification

When faced with a fraction, always start by simplifying it. In our example, \(\frac{5p^{8}}{100p^{13}}\), you have both numerical coefficients and variable terms.
Begin with separating the numbers from the variables. Look at \(\frac{5}{100}\) and \(\frac{p^{8}}{p^{13}}\).
This makes your job easier, as you can handle the numerical part and the variable part separately. Therefore, understand that simplifying fractions means reducing both numbers and variables individually.

greatest common divisor

To simplify a fraction like \(\frac{5}{100}\), find the Greatest Common Divisor (GCD) of the numerator and the denominator.
The GCD is the largest number that divides both without a remainder.
Here, the GCD of 5 and 100 is 5.
By dividing both the numerator and the denominator by their GCD, you get \(\frac{5 \div 5}{100 \div 5} = \frac{1}{20}\).
This dramatically simplifies your fraction effortlessly.

exponent properties

Exponents follow specific rules that make simplification easier. One key property for division is \(\frac{a^m}{a^n} = a^{m-n}\).
This rule helps to manage terms like \(\frac{p^8}{p^{13}}\) because you can simply subtract the exponents: \(p^{8-13} = p^{-5}\).
Notice that the exponent changes into a negative because you're taking a larger power away from a smaller power.

combining results

Once simplified, combine both your results to get the final answer.
From the previous steps, we have \(\frac{1}{20}\) from the numerical simplification and \(p^{-5}\) from the exponent properties.
Multiplying these together gives \(\frac{1}{20} \times p^{-5}\).
Since \(p^{-5}\) can be rewritten as \(\frac{1}{p^5}\), the final expression is \(\frac{1}{20p^5}\).
This is how you combine results to finish your simplification process.