Problem 8 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \frac{81 p^{6} q^{5}}{\left(3 p^{2} q\right)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(9p^2q^3\).

Step 1: Simplify the Denominator

Start by simplifying the expression in the denominator. The expression is \((3p^2q)^2\). When you have an expression in the form of \((a \times b \times c)^2\), you distribute the exponent to each term inside the parentheses. Thus, \((3p^2q)^2 = 3^2 \times (p^2)^2 \times q^2 = 9p^4q^2\).

Step 2: Set Up the Fraction

Now, place the original numerator over the simplified denominator: \[\frac{81 p^6 q^5}{9 p^4 q^2}\]

Step 3: Divide the Coefficients

Divide the coefficients (numerical parts) of the fraction. Divide 81 by 9: \[\frac{81}{9} = 9\]

Step 4: Divide the Variables

Apply the rules of exponents for division where you subtract the exponents of like bases: - For \(p\): \(p^6 \div p^4 = p^{6-4} = p^2\) - For \(q\): \(q^5 \div q^2 = q^{5-2} = q^3\) So the fraction simplifies to \(9p^2q^3\).

Step 5: Combine the Results

Combine the results from dividing the coefficients and variables. Therefore, the simplified expression is: \[9p^2q^3\]

Key Concepts

Exponents

Exponents are a way to express repeated multiplication. For example, instead of writing \( p \times p \times p \times p \), we use \( p^4 \), where 4 is the exponent indicating four occurrences of multiplication. Understanding exponents allows us to simplify expressions by handling large powers easily. In the exercise, we're dealing with raised powers such as \((3p^2q)^2\).

When raising a power to another power, we multiply the exponents. This principle simplifies expressions like \((p^2)^2\) to \(p^{2 \times 2} = p^4\). This is known as the "power of a power" rule. Similarly, for numerical bases, \(3^2\) becomes 9.

Exponents also play a significant role in manipulating algebraic expressions. They enable efficient division of variables by subtracting exponents of like bases, streamlining the simplification process.

Division of Monomials

Division of monomials involves dividing each part of a term separately. A monomial is a mathematical expression made up of a single term, like \(81p^6q^5\) or \(9p^4q^2\). When dividing monomials, we follow certain rules for both coefficients and variables.

• Coefficients: Simplify the numerical parts by straightforward division, such as \( \frac{81}{9} = 9\).
• Variables: Apply the division rule for exponents by subtracting exponents of like terms: \(p^6 \div p^4 = p^{6-4} = p^2\) and \(q^5 \div q^2 = q^{5-2} = q^3\).

The division of monomials allows us to break down complicated expressions into simpler forms by isolating and simplifying numbers and variables separately. By following these rules, we efficiently handle and simplify algebraic expressions.

Simplification Steps

Simplifying algebraic expressions involves a systematic approach to reduce complexity. In this exercise, we followed a series of steps:

• Simplify the Denominator: Break down expressions like \((3p^2q)^2\) using the power rules, resulting in \(9p^4q^2\).
• Set Up the Fraction: Place the original numerator (\(81p^6q^5\)) over the simplified denominator to create a fraction.
• Divide the Coefficients: Perform straightforward division on numbers, \( \frac{81}{9} \), to simplify to 9.
• Divide the Variables: Apply subtraction of exponents for each variable, as with \(p\) and \(q\).
• Combine the Results: Gather the results from previous steps to arrive at the simplified form, \(9p^2q^3\).

This approach breaks down the problem into manageable parts, ensuring clarity and simplicity throughout the process. By mastering these steps, one can efficiently simplify any similar algebraic expression.