Problem 20 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(\frac{3\left(x^{3}\right)^{4} y^{5}}{3 x^{7}}\)

Step-by-Step Solution

Answer

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

Step 1: Simplify the Numerator

The numerator of the expression is given as \( 3(x^3)^4 y^5 \). We need to simplify \((x^3)^4\) using the law of exponents \( (a^m)^n = a^{m imes n} \). So, \((x^3)^4 = x^{3 imes 4} = x^{12}\). Therefore, the numerator becomes \(3x^{12}y^5\).

Step 2: Simplify the Denominator

The denominator of the expression is \(3x^7\). Since there are no exponents to simplify in the denominator, we leave it as \(3x^7\).

Step 3: Divide the Expressions

The expression now is \(\frac{3x^{12}y^5}{3x^7}\). We can cancel the common factor \(3\) in the numerator and the denominator: \(\frac{3}{3} = 1\). Now the expression is \(\frac{x^{12}y^5}{x^7}\).

Step 4: Apply the Quotient Rule for Exponents

Use the quotient rule for exponents \(\frac{a^m}{a^n} = a^{m-n}\) to simplify \(\frac{x^{12}}{x^7}\). This results in \(x^{12-7} = x^5\). Thus, the expression simplifies to \(x^5y^5\).

Key Concepts

Laws of Exponents

Exponents, also known as powers, are mathematical expressions representing repeated multiplication of a number or a variable. The laws of exponents simplify expressions and make calculations more manageable. These laws include several key rules:

• Product of Powers: When multiplying two powers with the same base, add their exponents: \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Quotient of Powers: When dividing two powers with the same base, subtract their exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

In the exercise provided, we specifically utilized the power of a power rule to simplify \((x^3)^4\). By multiplying the exponents, \(x^{3 \times 4}\), we got \(x^{12}\). This step is crucial for reducing the expression to its simplest form.

Quotient Rule

The quotient rule for exponents helps simplify expressions involving division of powers with the same base. It states: \(\frac{a^m}{a^n} = a^{m-n}\). This rule is instrumental when you have a variable or number raised to a power in both the numerator and the denominator.

For instance, in the expression \(\frac{x^{12}}{x^7}\), both the numerator and the denominator have the base \(x\). By applying the quotient rule, we computed \(x^{12-7} = x^5\). Use this rule whenever bases in a fraction are identical, as it aids in simplifying complex algebraic expressions.

Numerator and Denominator Simplification

Simplifying numerators and denominators is fundamental in fraction operations. This simplification involves the cancellation of common factors and reducing complex expressions.

In our exercise, the first step involved simplifying the numerator. Initially, it was \(3(x^3)^4y^5\). By applying the laws of exponents, specifically the power of a power rule, we rewrote it as \(3x^{12}y^5\). Finding any common factors with the denominator, \(3x^7\), is the next step. Here, \(3\) was a common factor and could be canceled: \(\frac{3}{3}=1\). After cancelation, we were left with \(\frac{x^{12}y^5}{x^7}\).

This reduction through common factor cancelation simplifies expressions and clarifies the remaining calculation or simplification stages. Maintain consistency in searching for common elements, as reducing fractions often aids in further simplification.