Problem 31 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \frac{30 a^{-2} b^{-6}}{60 a^{-6} b^{-8}} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} a^4 b^2 \)

Step 1: Simplify the Coefficients

First, simplify the numerical coefficients. The expression has a coefficient of 30 in the numerator and 60 in the denominator. Divide both by their greatest common divisor, which is 30. Thus, \( \frac{30}{60} = \frac{1}{2} \).

Step 2: Use the Division Rule for Exponents on 'a'

Apply the division rule for exponents: \( \frac{a^m}{a^n} = a^{m-n} \). For the variable \( a \), we have \( \frac{a^{-2}}{a^{-6}} = a^{-2-(-6)} = a^{4} \).

Step 3: Use the Division Rule for Exponents on 'b'

Apply the division rule for exponents to the variable \( b \). This gives \( \frac{b^{-6}}{b^{-8}} = b^{-6-(-8)} = b^{2} \).

Step 4: Combine the Simplified Parts

Combine the results from the previous steps into one simplified fraction: \( \frac{1}{2} a^4 b^2 \).

Key Concepts

Exponents

Exponents are a vital mathematical tool to express repeated multiplication of a number by itself. For instance, if you have an exponent of 2 on a number, it means the number is multiplied by itself once, like this:

• \( n^2 = n \times n \)

In this exercise, exponents are used on variables such as 'a' and 'b'. The negative exponents \( a^{-2} \) and \( b^{-6} \) indicate that these variables are part of the denominator in the rational expression.
To simplify,

• Consider these negative exponents as taking the reciprocal of the base (making what's in the denominator move to the numerator, and vice-versa).
• The expression with a negative exponent can be changed to a positive exponent by inverting its position.

This property simplifies complex expressions and is part of why exponents are so powerful in algebra.

Division Rule for Exponents

When dividing expressions with the same base and exponents, the Division Rule for Exponents comes into play. This rule states:

• \( \frac{a^m}{a^n} = a^{m-n} \)

Here, the exponents are subtracted. When simplifying the expression involving 'a' and 'b', applying this rule is essential.

• For 'a', it transforms from \( a^{-2}/a^{-6} \) to \( a^{4} \).
• The same principle applies to 'b', \( b^{-6}/b^{-8} \) becomes \( b^{2} \).

This method ensures the exponents are handled correctly, leading to a simplified and more manageable expression.

Coefficient Simplification

In any algebraic expression involving coefficients, the simplification process begins with identifying the greatest common divisor (GCD) between the numerator and denominator.

• In the given problem, the coefficients in the fraction are 30 and 60.
• The GCD is 30.
• Dividing both coefficients by 30 results in \( \frac{30}{60} = \frac{1}{2} \).

By simplifying the coefficients first, you pave the way for a more straightforward manipulation of the variables and their exponents.
This step eliminates unnecessary complexity from the expression, leaving it in a sweeter form and preparing it for final simplification.