Problem 91 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{16 p^{3} q^{2}}{24 p q^{8}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2p^2}{3q^6} \).

Step 1: Factor the Coefficients

First, look at the coefficients in the expression \( \frac{16}{24} \). The greatest common factor of 16 and 24 is 8. Divide both the numerator and the denominator by 8: \[ \frac{16}{24} = \frac{16 \div 8}{24 \div 8} = \frac{2}{3}. \] So, the expression becomes \( \frac{2p^3 q^2}{3pq^8} \).

Step 2: Simplify the Powers of \( p \)

Next, simplify the powers of \( p \). In the expression \( \frac{2p^3}{3p} \), subtract the power in the denominator from the power in the numerator: \[ p^{3-1} = p^2. \] Thus, we now have \( \frac{2p^2 q^2}{3q^8} \).

Step 3: Simplify the Powers of \( q \)

Now, simplify the powers of \( q \) in \( \frac{2q^2}{q^8} \). Subtract the power in the denominator from the power in the numerator: \[ q^{2-8} = q^{-6}. \] The expression becomes \( \frac{2p^2}{3q^6} \).

Step 4: Final Expression

Combine the results from the previous steps. The simplified expression is \( \frac{2p^2}{3q^6} \).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is an essential concept in algebraic simplification. It refers to the largest number that can evenly divide two or more numbers. When simplifying expressions with coefficients, finding the GCF of the coefficients allows you to reduce the expression to its simplest form.

• To determine the GCF, look at the numbers' prime factorizations and identify the largest factor common to all numbers.
• For example, with the coefficients 16 and 24, the prime factorizations are: 16 = 2 × 2 × 2 × 2 and 24 = 2 × 2 × 2 × 3. The common factors are three 2's, so the GCF is 2 × 2 × 2 = 8.

Using the GCF, you can divide both the numerator and denominator by this factor, making the expression simpler. For instance, dividing both 16 and 24 by their GCF, 8, simplifies the fraction to \( \frac{2}{3} \). This step prepares the expression for further simplification of variables.

Exponents

Exponents indicate how many times a number or variable is multiplied by itself. Understanding exponents is crucial when simplifying expressions with variables each raised to certain powers. Here's a simple way to handle exponents, especially when you are dividing:

• When dividing variables with the same base, subtract the exponent in the denominator from the exponent in the numerator.

For example, in the expression \( \frac{p^3}{p} \), subtract 1 from 3 to simplify \( p^{3-1} = p^2 \).This simplification rule can be applied to any variables with exponents. Be careful not to mistakenly multiply the bases when you should be simplifying in division.
Remember that a negative exponent means the reciprocal of the positive exponent. For instance, \( q^{-6} \) is equivalent to \( \frac{1}{q^6} \), showing how negative exponents move variables to the denominator.

Ratio of Polynomials

A ratio of polynomials refers to a fraction where both the numerator and the denominator are polynomials. Simplifying these ratios involves several steps, including factoring and reducing terms by common factors.To simplify a ratio of polynomials:

• First, simplify any coefficients using their greatest common factor, as we did with \( \frac{16}{24} \) by dividing each by 8 to get \( \frac{2}{3} \).
• Next, look at the variables with exponents. In a polynomial ratio like \( \frac{p^3q^2}{pq^8} \), you subtract corresponding exponents.
• After simplifying both the coefficients and variables, the expression turns into its simplest form, such as \( \frac{2p^2}{3q^6} \).

Understanding and simplifying the ratio of polynomials can greatly reduce complex expressions, making it easier to work with them in further mathematical problems. It's all about systematically breaking down each part and ensuring every component is in its simplest form.