Problem 92 - Intermediate Algebra

Question

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{30 a^{3} b^{15}}{18 a^{9} b^{10}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{5b^{5}}{3a^{6}} \)."

Step 1: Identify the common factors in the numerator and denominator

The given expression is \( \frac{30 a^{3} b^{15}}{18 a^{9} b^{10}} \). Identify the greatest common factor (GCF) for the coefficients 30 and 18, which is 6. For the variables, simplify using the properties of exponents: subtract the exponent in the denominator from the exponent in the numerator for each variable.

Step 2: Simplify the coefficients

Divide the coefficients by their greatest common factor: \( \frac{30}{18} = \frac{5}{3} \). This simplifies the numerical part of the expression.

Step 3: Simplify the variable with base 'a'

For the variable 'a', subtract the exponent in the denominator from the exponent in the numerator: \( a^{3-9} = a^{-6} \). This expresses the variable with a negative exponent, which can be rewritten as \( \frac{1}{a^{6}} \).

Step 4: Simplify the variable with base 'b'

For the variable 'b', subtract the exponent in the denominator from the exponent in the numerator: \( b^{15-10} = b^{5} \).

Step 5: Write the simplified expression

Combine the simplified parts: coefficients \( \frac{5}{3} \), \( b^{5} \) in the numerator, and \( a^{6} \) in the denominator. The expression simplifies to \( \frac{5b^{5}}{3a^{6}} \).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a fundamental concept in mathematics, particularly useful in simplifying algebraic expressions and fractions. It refers to the largest number that divides two or more numbers without leaving a remainder. When simplifying algebraic fractions, the GCF helps reduce the coefficients to their simplest form.

For instance, in our exercise with the expression \( \frac{30 a^{3} b^{15}}{18 a^{9} b^{10}} \), the coefficients 30 and 18 must be reduced. The GCF of 30 and 18 is 6 because:

• The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
• The factors of 18 are 1, 2, 3, 6, 9, 18.
• The highest common factor is 6.

By dividing the coefficients 30 and 18 by their GCF (6), the expression becomes simplified in terms of the numerical part: \( \frac{5}{3} \). This step is crucial as it lays the groundwork for simplifying the entire algebraic fraction effectively.

Properties of Exponents

Exponents are a powerful tool in mathematics, allowing us to efficiently represent repeated multiplication. Simplifying algebraic expressions often involves understanding and applying the properties of exponents correctly.

• One key property is when multiplying two powers with the same base, you add their exponents: \( a^{m} \cdot a^{n} = a^{m+n} \).
• When dividing, you subtract the exponents: \( \frac{a^{m}}{a^{n}} = a^{m-n} \).

In the given problem, this property helps us manage the exponents of the variables 'a' and 'b'. We subtract exponents in the denominator from those in the numerator:

• For 'a': \( a^{3-9} = a^{-6} \).
• For 'b': \( b^{15-10} = b^{5} \).

Using these properties not only simplifies the expression but also paves the way for understanding other complex algebraic operations. This subtraction might result in negative exponents, which leads us to our next important concept.

Negative Exponents

Negative exponents may seem intimidating, but they are simply a way to express reciprocals. When you encounter a negative exponent, it indicates the reciprocal of the base raised to the corresponding positive exponent. Simply put:

• \( a^{-n} = \frac{1}{a^{n}} \).

In the context of our expression, after applying the properties of exponents, we get \( a^{-6} \). This can be rewritten as \( \frac{1}{a^{6}} \), effectively moving the base 'a' from the numerator to the denominator.

It’s essential to realize that moving bases across the fraction line changes the sign of the exponent. Mastering negative exponents allows you to transition smoothly between different forms of expressions, enhancing your overall capability in algebraic manipulation.