Problem 78 - Algebra 2

Question

PREREQUISITE SKILL. Simplify. Assume that no variable equals 0. $$ \frac{9 a^{3} b}{3 a b} $$

Step-by-Step Solution

Answer

\(3a^2\)

Step 1: Break Down the Expression

First, let's break down the components of the expression \( \frac{9a^3b}{3ab} \). The numerator is \( 9a^3b \) and the denominator is \( 3ab \). Each part consists of numbers and variables that can be simplified from the numerator to the denominator.

Step 2: Simplify Numerical Coefficients

Focus on simplifying the numerical coefficients. Divide the coefficient 9 in the numerator by the coefficient 3 in the denominator: \( \frac{9}{3} = 3 \). So the expression becomes \( \frac{3a^3b}{ab} \).

Step 3: Simplify the Variables

Now simplify the variables by canceling common terms in the numerator and the denominator. The variable \( a \) appears as \( a^3 \) in the numerator and \( a^1 \) in the denominator. Divide \( a^3 \) by \( a \) to get \( a^{3-1} = a^2 \). Similarly, the variable \( b \) in the numerator and denominator cancels out completely since \( \frac{b}{b} = 1 \). The expression simplifies to \( 3a^2 \).

Step 4: Final Expression

After simplifying the coefficients and variables, the final simplified expression is \( 3a^2 \).

Key Concepts

Numerical Coefficients

In algebraic expressions, numerical coefficients are simply the numbers that multiply the variables. In our given problem, the expression starts as \( \frac{9a^3b}{3ab} \). Here, the coefficients 9 and 3 are the numbers directly placed before the terms containing the variables. To simplify numerical coefficients, we perform a division operation. We divide the coefficient in the numerator by the one in the denominator:

• The numerator has a coefficient of 9.
• The denominator has a coefficient of 3.

When we divide 9 by 3, the result is 3. Thus, we have simplified the expression to \( \frac{3a^3b}{ab} \). This shows the importance of handling numerical coefficients properly, as it sets the foundation for simplifying the whole expression.

Variable Cancellation

Variable cancellation is crucial when simplifying algebraic fractions. It occurs when the same variable is present in both the numerator and the denominator, allowing us to "cancel" them out.In our expression \( \frac{3a^3b}{ab} \), both the numerator and denominator share the variables \( a \) and \( b \). Here's how they cancel:

• The variable \( a \) appears as \( a^3 \) in the numerator and as \( a^1 \) in the denominator. By dividing them, we perform \( a^{3-1} = a^2 \).
• The \( b \) variables completely cancel each other out because \( \frac{b}{b} = 1 \).

After canceling, only \( a^2 \) remains, leading to the simplified form of the expression: \( 3a^2 \). Understanding how variable cancellation works helps in simplifying complex algebraic expressions efficiently and accurately.

Exponents

Exponents play a significant role in algebraic simplification. They indicate the power to which a number or variable is raised. For example, in our expression \( a^3 \) means that the variable \( a \) is multiplied by itself three times.Simplifying exponents involves reducing the power by canceling or combining.

• When dividing terms, we subtract the exponents of the same bases: \( a^3 \div a^1 = a^{3-1} = a^2 \).

This operation effectively reduces the power of the variable when both numerator and denominator share the same base, streamlining the expression. It is essential to carefully consider exponents when simplifying, as they can significantly alter the outcome if not handled correctly.

Fractions in Algebra

Handling fractions is a fundamental skill in algebra. They often involve terms made up of both numbers and variables. To simplify fractions in algebra, like \( \frac{9a^3b}{3ab} \), it's important to break down both parts into their simplest forms.Fractions are simplified by canceling out common factors in both the numerator and the denominator.

• Start by simplifying numerical coefficients by division.
• Next, cancel out any common variables by reducing powers, as done with the \( a^3 \) and \( b \) in our example.

By performing these operations, fractions can be condensed into a simpler form, such as turning our given expression into \( 3a^2 \). This ensures the fraction is as concise and understandable as possible, facilitating further algebraic operations.