Problem 5 - Elementary Algebra

Question

For exercises 1-66, simplify. $$ \frac{90 n^{2} p^{8}}{42 n^{5} p^{6}} $$

Step-by-Step Solution

Answer

\[ \frac{15 p^{2}}{7 n^{3}} \]

Step 1: Write Down the Problem

Start by writing down the given expression: \[ \frac{90 n^{2} p^{8}}{42 n^{5} p^{6}} \]

Step 2: Simplify the Coefficients

Simplify the coefficients 90 and 42 by dividing both by their greatest common divisor (GCD), which is 6. \[ \frac{90}{42} = \frac{90 \times \frac{1}{6}}{42 \times \frac{1}{6}} = \frac{15}{7} \] So, the expression becomes: \[ \frac{15 n^{2} p^{8}}{7 n^{5} p^{6}} \]

Step 3: Simplify the Variables Individually

Simplify the variable terms by subtracting the exponents in the denominator from the exponents in the numerator for both variables n and p. \[ \text{For } n: n^{2-5} = n^{-3} = \frac{1}{n^{3}} \] \[ \text{For } p: p^{8-6} = p^{2} \] So, the expression becomes: \[ \frac{15 p^{2}}{7 n^{3}} \]

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial concept when simplifying fractions, including algebraic ones. It refers to the largest number that can divide two (or more) numbers without leaving a remainder. In the context of the problem, we look at the coefficients 90 and 42.

To find the GCD:
1. List the factors of each number.
2. Identify the largest common factor.

For 90, factors are: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.
For 42, factors are: 1, 2, 3, 6, 7, 14, 21, 42.

The common factors are 1, 2, 3, and 6. Thus, the GCD is 6.

Dividing the coefficients by their GCD simplifies the expression, making complex equations more manageable and easier to solve.

Exponent Rules

Exponents, or powers, represent repeated multiplication of a number by itself. In simplifying algebraic expressions, understanding and applying exponent rules is fundamental. The essential rule used here is the division of powers with the same base.
When you have the same base being divided, you simply subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

Applying this to our problem:
\(n^2\) in the numerator and \(n^5\) in the denominator simplifies as follows:
\

Fraction Simplification

Simplifying fractions is at the heart of this algebra problem. It involves reducing a fraction to its simplest form without changing its value. A fraction is fully simplified when the numerator and denominator have no common factors other than 1.

Steps to simplify a fraction:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• Apply the exponent rules to simplify variable expressions.

In our problem, we simplify the algebraic expression \(\frac{90 n^2 p^8}{42 n^5 p^6}\):

1. Finding the GCD of 90 and 42, which is 6.
2. Dividing both terms by 6 yields \(\frac{15 n^2 p^8}{7 n^5 p^6}\).
3. Applying exponent rules: for \`n\`, \`n^{2-5} = n^{-3}\` which simplifies to \( \frac{1}{n^3} \); for \`p\`, \`p^{8-6} = p^2\`.

Thus, the simplified expression is \(\frac{15 p^2}{7 n^3}\).