Problem 38 - Algebra 2

Question

Simplify each expression. \(\frac{\left(-3 x^{2} y\right)^{3}}{9 x^{2} y^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \\(-3x^4y\\).

Step 1: Expand the Numerator

First, express the numerator in expanded form. The expression \((-3x^2y)^3\) represents the cube of each component within the parentheses. This means expanding it to: \( (-3)^3(x^2)^3(y)^3 \). Calculating each part gives us \((-3)^3 = -27\), \(x^{2\times3} = x^6\), and \(y^3 = y^3\). Thus, the expanded form is \(-27x^6y^3\).

Step 2: Write the Full Expression

Now substitute the expanded numerator back into the original fraction. The expression becomes: \(\frac{-27x^6y^3}{9x^2y^2}\).

Step 3: Simplify the Fraction

Next, simplify the fraction by dividing both the numerator and the denominator by the common factors. Start with the numerical part: \(-27\div9 = -3\). Then, apply the quotient rule for exponents to simplify \(x^6\div x^2 = x^{6-2}=x^4\) and \(y^3\div y^2 = y^{3-2}=y\). This simplifies the expression to \(-3x^4y\).

Step 4: Verify the Simplification

Recheck each part of the simplification to ensure that all calculations are correct. \((-3)^3 = -27\), \(x^{6-2}=x^4\), \(y^{3-2}=y\). The simplified expression is \(-3x^4y\).

Key Concepts

Exponents

Exponents are a fundamental concept in algebra that help us to express repeated multiplication of a number by itself. For instance, if we have the expression \(a^b\), it means that the number \(a\) is multiplied by itself \(b\) times. Exponents allow us to rewrite long multiplication problems in a more compact form, making calculations easier to manage.
This is especially useful in algebraic expressions and equations where the same variable is repeated multiple times.Some important rules regarding exponents include:

• Multiplicative Law: \(a^m \times a^n = a^{m+n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Division with Same Base: \(a^m \div a^n = a^{m-n}\)
• Exponent of One: \(a^1 = a\)
• Exponent of Zero: \(a^0 = 1\), provided \(a eq 0\)

These rules are vital when working with exponents as they allow us to simplify expressions and solve algebraic problems efficiently.

Polynomial Simplification

Polynomial simplification is the process of reducing a polynomial to its simplest form, eliminating any like terms, and arranging the terms in an order that makes computation easier. Simplifying polynomials involves applying several algebraic rules and properties to condense the expression. When simplifying polynomials, it is essential to

• Combine like terms: Terms with the exact variable and exponent.
• Expand terms: Applying laws of exponents to handle powers of a power.
• Factor out common factors: Simplifying the polynomial by removing shared factors from terms if applicable.

In the given problem, simplification is made by expanding the terms and thoroughly applying the rules to exponents, handling each term separately and ensuring no calculation errors.
This allows us to easily eliminate complex fractions and simplify calculations in further steps.

Quotient Rule for Exponents

The quotient rule for exponents is a crucial algebraic tool that helps simplify expressions where terms with the same base are divided by each other. This rule states that when dividing two exponents with the same base, you should subtract their exponents:\[ a^m \div a^n = a^{m-n} \]
In our original expression \( \frac{-27x^6y^3}{9x^2y^2} \), we see that the quotient rule is heavily used. For the variable \(x\), the exponents \(6\) and \(2\) are subtracted to simplify to \(x^4\). For the variable \(y\), the exponents \(3\) and \(2\) work similarly to simplify to \(y\).Understanding the quotient rule is essential for efficiently breaking down and simplifying complex algebraic expressions. It allows a seamless transition from expanded to reduced polynomial forms by handling both numerical and variable exponents consistently.