Problem 70 - Algebra for College Students

Question

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\frac{108 a^{-5} b^{-4}}{9 a^{-2} b}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{12}{a^3 b^5} \).

Step 1: Simplify the Coefficients

Start by simplifying the numerical coefficient in the expression: \( \frac{108}{9} = 12 \).

Step 2: Apply the Quotient Rule for Exponents

The quotient rule states that \( \frac{a^m}{a^n} = a^{m-n} \). Apply this to the variable \( a \): \( \frac{a^{-5}}{a^{-2}} = a^{-5-(-2)} = a^{-3} \).

Step 3: Simplify Variable \( b \) Using the Quotient Rule

Use the quotient rule for the variable \( b \): \( \frac{b^{-4}}{b^1} = b^{-4-1} = b^{-5} \).

Step 4: Convert Negative Exponents to Positive Exponents

Rewrite \( a^{-3} \) and \( b^{-5} \) with positive exponents: \( a^{-3} = \frac{1}{a^3} \) and \( b^{-5} = \frac{1}{b^5} \).

Step 5: Combine Intermediate Results

Combine all parts: the coefficient 12 and the reciprocals of \( a^3 \) and \( b^5 \). This gives \( \frac{12}{a^3 b^5} \).

Key Concepts

Quotient Rule for Exponents

Understanding the Quotient Rule for exponents is crucial in simplifying algebraic expressions. The rule helps us deal with expressions where the same base appears in both the numerator and the denominator of a fraction. According to this rule, when you divide like bases, you simply subtract the exponent in the denominator from the exponent in the numerator.
This is expressed mathematically as: if you have \( \frac{a^m}{a^n} \), it simplifies to \( a^{m-n} \). Let's break this down with a simple example:

• Imagine you have \( \frac{x^5}{x^3} \). By applying the quotient rule, this simplifies to \( x^{5-3} = x^2 \).
• Even when exponents are negative, such as \( \frac{x^{-5}}{x^{-2}} \), the rule still applies, resulting in \( x^{-5-(-2)} = x^{-3} \).

When using the Quotient Rule, always remember to subtract carefully to avoid errors.

Negative Exponents

Negative exponents often confuse students, but don't worry—it's more straightforward than it seems! Essentially, a negative exponent indicates the reciprocal of the base raised to the opposite positive exponent.
For example, \( a^{-n} \) can be rewritten as \( \frac{1}{a^n} \). This means that any base with a negative exponent is equivalent to the reciprocal of the base with a positive exponent.
This rule helps simplify expressions like:

• \( x^{-3} \) becomes \( \frac{1}{x^3} \).
• Similarly, \( \frac{y^{-5}}{y^{-2}} = y^{-5+2} = y^{-3} \) simplifies further to \( \frac{1}{y^3} \).

Understanding negative exponents is essential for rewriting expressions in ways that are easier to understand and manipulate.

Positive Integral Exponents

Positive integral exponents signify repeated multiplication of the base. For example, \( a^3 \) represents \( a \times a \times a \). Simply put, any positive integer exponent shows how many times the base is multiplied by itself.
When simplifying expressions, it's always ideal to convert any negative exponents into positive ones. This is done to follow standard mathematical convention, which often requires solutions in a positive integral form.
Converting any expression ensures it's easier to interpret and more consistent across mathematical problems. For instance:

• \( b^{-4} \) converts to \( \frac{1}{b^4} \), which is a positive integral exponent form.
• In an expression like \( \frac{12}{a^3 b^5} \), all exponents are positive, meeting the required standard.

Thus, working with positive integral exponents makes the final expression neat and universally recognizable.

Simplifying Coefficients

Simplifying coefficients refers to reducing the numerical part of an algebraic expression to its simplest form. This is an essential step since it simplifies calculations and helps in understanding the core of the expression.
In the given problem, we started with \( \frac{108}{9} \), simplifying it to \( 12 \). Coefficients, simply put, are the numerical factors that multiply the variables in an expression.
Reducing coefficients involves basic arithmetic:

• You might divide, multiply, or further simplify fractions.
• In \( \frac{108}{9} \), dividing 108 by 9 gives a neat and simpler coefficient of 12.

By breaking down coefficients to their simplest form, not only is the expression more manageable, but it also sets a solid foundation for further algebraic manipulation.