Problem 3 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \frac{-2 a^{3} b^{6}}{18 a^{2} b^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-a b^{4}}{9} \).

Step 1: Simplify the Coefficient

The numerator coefficient is -2 and the denominator coefficient is 18. To simplify these, find the greatest common divisor (GCD) of 2 and 18, which is 2. Divide both by 2: \( \frac{-2}{18} = \frac{-1}{9} \).

Step 2: Simplify the Power of 'a'

The powers of \( a \) are 3 in the numerator and 2 in the denominator. To simplify, subtract the exponent in the denominator from the exponent in the numerator: \( a^{3-2} = a^{1} \).

Step 3: Simplify the Power of 'b'

The powers of \( b \) are 6 in the numerator and 2 in the denominator. To simplify, subtract the exponent in the denominator from the exponent in the numerator: \( b^{6-2} = b^{4} \).

Step 4: Write the Final Expression

Combine the simplified components from the previous steps into one fraction: \( \frac{-1}{9} a^{1} b^{4} \). This simplifies to \( \frac{-a b^{4}}{9} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is a key process in algebra that involves reducing a fraction to its simplest form. This means making the numerator and denominator as small as possible while maintaining the same value.
To simplify, first identify any common factors between the numerator and the denominator. It’s much like sharing a pizza equally; you find what they both can be divided by. For example:

• For the fraction \( \frac{-2}{18} \), the greatest common factor is 2.
• Dividing the numerator and denominator by 2 simplifies it to \( \frac{-1}{9} \).

Simplifying is not just for numbers but also for expressions with variables, as factors can include both numbers and variables.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide two or more numbers without leaving a remainder. Understanding the GCD is crucial when simplifying fractions because it helps reduce the numbers to their simplest form.
To find the GCD:

• List all the divisors of each number, or use prime factorization.
• For the numbers 2 and 18 in our example, the divisors are 1, 2, and 1, 2, 3, 6, 9, 18 respectively.
• The largest divisor they share is 2, making it the GCD.

By dividing both parts of the fraction \( \frac{-2}{18} \) by 2, you can simplify it. This principle applies equally when dealing with algebraic expressions too.

Exponents

Exponents are a way to express repeated multiplication of the same number or variable. They are written as a small number above and to the right of the base number or letter.
Understanding how to manipulate exponents is essential, especially when simplifying expressions. For example:

• In \( a^3 \), \( a \) is multiplied by itself three times.
• When dividing powers with the same base, subtract the exponents: \( a^3 / a^2 = a^{3-2} = a^1 \).
• In the exercise, subtracting exponents simplifies \( b^6 / b^2 \) to \( b^{6-2} = b^4 \).

By using these properties, expressions with exponents can be simplified significantly.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are the backbone of algebra, allowing us to describe mathematical relationships concisely.
An expression may involve "+", "-", "*", or "/" operations and can be simplified by applying arithmetic operations along with the rules of algebra. For instance:

• The expression \( \frac{-2 a^{3} b^{6}}{18 a^{2} b^{2}} \) combines coefficients and variables with exponents.
• To simplify, first handle the numeric coefficients, followed by variables using exponent rules.
• Ultimately, it reduces to \( \frac{-a b^4}{9} \), representing a much simpler form.

In algebra, simplifying makes expressions easier to understand and solve in further calculations.