Problem 52 - College Algebra with Modeling and Visualization

Question

Exercises \(45-54:\) Use the quotient rule to simplify the expression. Use positive exponents to write your answer. $$ \frac{-6 x^{7} y^{3}}{3 x^{2} y^{-5}} $$

Step-by-Step Solution

Answer

The simplified expression is \(-2x^5y^8\).

Step 1: Identify the Quotient Rule

The quotient rule for exponents states that when you divide two powers with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \). We'll apply this rule to each variable in the given expression.

Step 2: Simplify the Coefficients

Divide the coefficients \(-6\) and \(3\):\[ \frac{-6}{3} = -2 \].So the simplified coefficient of the expression is \(-2\).

Step 3: Simplify the Exponents of \(x\)

Apply the quotient rule to the variable \(x\):\[ \frac{x^7}{x^2} = x^{7-2} = x^5 \].This means that the exponent of \(x\) in the simplified expression is \(5\).

Step 4: Simplify the Exponents of \(y\)

Apply the quotient rule to the variable \(y\):\[ \frac{y^3}{y^{-5}} = y^{3 - (-5)} = y^{3+5} = y^8 \].This means that the exponent of \(y\) in the simplified expression is \(8\).

Step 5: Write the Final Expression

Combine the simplified components to get the final expression:\[ -2x^5y^8 \].This expression uses positive exponents as required.

Key Concepts

Understanding Exponents

Exponents are a way to represent repeated multiplication of a number by itself. When you see a number like \(x^7\), it means that \(x\) is multiplied by itself 7 times: \(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x\). The laws of exponents make it easier to simplify expressions involving multiplication and division of numbers with the same base. Here we use the quotient rule, one of these important exponent rules.

• Product of Powers: To multiply powers with the same base, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power: To raise a power to another power, multiply the exponents: \((a^m)^n = a^{mn}\).
• Quotient of Powers: To divide powers with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

Understanding these rules can help simplify complex mathematical expressions efficiently.

Simplifying Expressions

Simplifying an expression means transforming it into its most reduced form without changing its overall value. This is essential in algebra as it helps make expressions easier to understand and work with. In the example given, we start by simplifying the coefficients before applying the rules of exponents. The operation \(\frac{-6}{3} = -2\) simplifies the numerical part of the expression. Now, apply the quotient rule of exponents to each variable separately.

• Simplifying Coefficients: Always simplify the numbers in front of the variables first. Perform basic arithmetic operations such as division or multiplication.
• Applying Quotient Rule: Follow by dealing with variables. For \(x\), subtract the exponent in the denominator \(x^2\) from that in the numerator \(x^7\) resulting in \(x^5\). For \(y\), \(y^{-5}\) in the denominator becomes \(y^5\) when subtracted from \(y^3\), resulting in \(y^8\).

These steps condense the expression into its simplest form, which is easier to understand and use in further calculations.

Understanding Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both are algebraic expressions. Simplifying algebraic fractions can involve the rules of arithmetic combined with algebraic principles like factoring and exponent rules. The primary goal when working with algebraic fractions is to simplify them, just like numerical fractions. Here are key techniques:

• Factor Both Numerator and Denominator: Whenever possible, factor the algebraic expressions to cancel common factors.
• Apply Exponent Rules: Simplify variables in the fraction using rules like the quotient rule for exponents.
• Positive Exponents: Always express the final expression using positive exponents, unless otherwise specified.

In our example, we simplify the algebraic fraction by dividing the coefficients and applying the quotient rule to variables separately. By understanding these techniques, handling algebraic fractions becomes an easier and more straightforward task.