Problem 53 - Intermediate Algebra

Question

Divide, and then simplify, if possible. See Example 6. $$ \frac{p q^{29}}{50} \div \frac{p^{10} q^{38}}{15} $$

Step-by-Step Solution

Answer

\( \frac{3}{10p^9 q^9} \)

Step 1: Rewrite the Division as Multiplication

When we divide by a fraction, we multiply by its reciprocal. Rewrite the expression: \( \frac{p q^{29}}{50} \div \frac{p^{10} q^{38}}{15} \) becomes \( \frac{p q^{29}}{50} \times \frac{15}{p^{10} q^{38}} \).

Step 2: Simplify the Numerical Part

Multiply the fractions: \( \frac{p q^{29} \times 15}{50 \times p^{10} q^{38}} \). Simplify the coefficients: \( \frac{15}{50} = \frac{3}{10} \). Now the expression is: \( \frac{3p q^{29}}{10 p^{10} q^{38}} \).

Step 3: Simplify the Variables using Laws of Exponents

Use the law of exponents \( \frac{x^m}{x^n} = x^{m-n} \). For \( p \), \( \frac{p^1}{p^{10}} = p^{1-10} = p^{-9} \). For \( q \), \( \frac{q^{29}}{q^{38}} = q^{29-38} = q^{-9} \). Rewrite the expression: \( \frac{3}{10}p^{-9}q^{-9} \).

Step 4: Rewrite with Positive Exponents

Convert negative exponents to positive by writing them as reciprocals: \( \frac{3}{10} \times \frac{1}{p^9 q^9} = \frac{3}{10p^9 q^9} \). Thus, the simplified expression is: \( \frac{3}{10p^9 q^9} \).

Key Concepts

Laws of Exponents

Understanding the Laws of Exponents is crucial when working with algebraic fractions. Exponents represent repeated multiplication. The main rule used here is the quotient of powers rule, which states:

• \( \frac{x^m}{x^n} = x^{m-n} \)

This is useful when simplifying expressions with the same base. We subtract the exponent in the denominator from the exponent in the numerator. This rule helps streamline complex expressions by reducing them to a simpler form, making calculations easier.
In our exercise, the laws of exponents help us simplify terms like \( \frac{q^{29}}{q^{38}} \). It becomes \( q^{29-38} = q^{-9} \), illustrating how the rule transforms expressions. Remember, if the result gives a negative exponent, it can be converted to a positive exponent by writing it as a reciprocal. This is seen when \( p^{-9} \) and \( q^{-9} \) are expressed as \( \frac{1}{p^9} \) and \( \frac{1}{q^9} \). This conversion ensures all exponents are positive, maintaining a neat and clear final expression.

Fraction Division

Division of fractions can initially seem daunting, but it's straightforward once you get the hang of the process. The key is to convert division into multiplication. This can be done by multiplying by the reciprocal of the fraction you are dividing by.

To clarify:

• The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).
• Therefore, to divide by \( \frac{p^{10} q^{38}}{15} \), you multiply by its reciprocal \( \frac{15}{p^{10} q^{38}} \).

This conversion is useful because it allows you to manage the division operation by repeating a simpler multiplication operation. It simplifies juggling multi-step fractions and focuses instead on applying familiar multiplication methods.
Once converted, multiplying the numerators together and the denominators together simplifies visual representation of the problem. This is a vital first step to solving complex algebraic fraction problems.

Simplifying Expressions

After converting division to multiplication and applying the laws of exponents, the next step is simplifying the entire expression. Simplification is about reducing an expression to its simplest form, stripping it of any unnecessary complexity.
Firstly, handle any numerical coefficients by simplifying the fraction formed by the coefficients. For instance, the fraction \( \frac{15}{50} \) simplifies to \( \frac{3}{10} \). This step is critical because it makes the expression cleaner and easier to manage.

• Look at variable expressions and apply exponent rules.
• For negative exponents, remember to convert them to positive through inversion.

So, terms like \( p^{-9} \) and \( q^{-9} \) are converted to \( \frac{1}{p^9} \) and so on. By rewriting negative exponents with positive ones, we make expressions more aesthetically pleasing and mathematically correct.
Ultimately, by breaking down and simplifying each component—numerical and variable—the expression \( \frac{3}{10}p^{-9}q^{-9} \) turns into \( \frac{3}{10p^9 q^9} \), which is the final, simplified result where each factor is clearly defined.