Problem 63 - Algebra for College Students

Question

Find each quotient. $$\frac{-54 a b^{2} c^{3}}{-6 a b c}$$

Step-by-Step Solution

Answer

9bc^2

Step 1: Simplify the Numerical Coefficients

First, take the numerical coefficients of the given expression: \(-54\) and \(-6\). Divide \(-54\) by \(-6\): \[ \frac{-54}{-6} = 9 \] This gives us a numerical coefficient of 9 for the simplified expression.

Step 2: Simplify the Variables

Now, simplify each variable separately: - For \(a\): Cancel out the \(a\) from the numerator and the denominator since both are \(a^1\).- For \(b\): Subtract the exponent in the denominator from the exponent in the numerator: \[ b^{2-1} = b^1 = b \]- For \(c\): Subtract the exponent in the denominator from the exponent in the numerator: \[ c^{3-1} = c^2 \]

Step 3: Combine the Simplified Terms

Combine all the simplified parts together. From Step 1, we obtained a numerical coefficient of 9. From Step 2, the simplified variables are \(b\) and \(c^2\).Therefore, the expression can be combined into:\[ 9bc^2 \]

Key Concepts

Numerical Coefficients Simplification

Simplifying numerical coefficients is an essential step in simplifying algebraic expressions. In the given expression \( \frac{-54 a b^{2} c^{3}}{-6 a b c} \), we first focus on the numbers: \(-54\) and \(-6\).

To simplify, divide the numerator by the denominator:\[-54 \div -6 = 9\].

This division essentially strips away any unnecessary values and isolates what remains important for the expression, which in this case is the coefficient 9. By focusing on numerical coefficients first, you simplify the arithmetic early, making subsequent steps easier.

Variable Cancellation

Variable cancellation occurs when the same variable appears in both the numerator and the denominator. In our example, each variable should be investigated individually.

For the variable \(a\), both numerator and denominator contain \(a^1\). Since these can cancel each other out completely, we're left with no \(a\) in the simplified quotient.

When a variable in the numerator and denominator shares the same exponent, it simplifies to one because anything divided by itself equals one. Thus, understanding variable cancellation can help streamline complex expressions.

Exponent Subtraction

Exponent subtraction is a core concept used to simplify expressions with variables across fractions. In the expression \( \frac{-54 a b^{2} c^{3}}{-6 a b c} \), after cancelling a variable that doesn’t appear more than once, we move on to variables \(b\) and \(c\), which differ in exponents.

For \(b\), subtract the exponent in the denominator (\(1\)) from the exponent in the numerator (\(2\)), resulting in \(b^{2-1} = b^1 = b\).

For \(c\), subtract the denominator's exponent from the numerator's: \(c^{3-1} = c^2\). This approach allows for reducing expressions by observing that powers of the same base divide by subtracting exponents.

Algebraic Expression Simplification

Simplifying algebraic expressions can appear daunting, but becomes manageable with clear strategies. For the initially complex expression \( \frac{-54 a b^{2} c^{3}}{-6 a b c} \), simplification is achieved by breaking it down.

Start with numerical simplification to tackle coefficients first, reducing initial confusion as seen with \(-54 \div -6 = 9\). Proceed to variables: cancel those totally similar, and for differing exponents, use subtraction methods.

Lastly, combine all simplified elements—all leading to the consolidated expression \(9bc^2\). This process transforms unwieldy expressions into comprehensible, simple terms.