Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \frac{\left(6 x^{3}\right)^{2}}{\left(2 x^{2}\right)^{3}} \cdot\left(3 x^{2}\right)^{0} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{9}{2}\).

Step 1: Simplify Exponents

The term \((3x^{2})^{0}\) is raised to the power of 0. Any non-zero number raised to the power of 0 is 1. Thus, we have \((3x^{2})^{0} = 1\). The expression now becomes \(\frac{(6x^{3})^{2}}{(2x^{2})^{3}} \cdot 1\).

Step 2: Apply Power to Each Term

Apply power of a power rule to \((6x^{3})^{2}\) and \((2x^{2})^{3}\). This means raising every component inside of each parentheses to the given power. So, \((6x^{3})^{2}\) becomes \(6^{2}x^{6}\) (since \(x^{3 \cdot 2} = x^{6}\)), and \((2x^{2})^{3}\) becomes \(2^{3}x^{6}\) (since \(x^{2 \cdot 3} = x^{6}\)). The expression now is \(\frac{36x^{6}}{8x^{6}}\).

Step 3: Simplify the Fraction

The expression \(\frac{36x^{6}}{8x^{6}}\) can be simplified by canceling \(x^{6}\) on both the numerator and the denominator, resulting in \(\frac{36}{8}\).

Step 4: Reduce the Fraction

Simplify the fraction \(\frac{36}{8}\) by dividing the numerator and the denominator by their greatest common divisor, which is 4. Thus, \(\frac{36}{8} = \frac{36 \div 4}{8 \div 4} = \frac{9}{2}\).

Key Concepts

Exponent Rules

Exponent rules are fundamental for simplifying algebraic expressions, especially when dealing with powers. The rules dictate how to handle expressions involving exponents, ensuring calculations are accurate and simplified effectively. There are several key exponent rules, including:

• Product of Powers Rule: When multiplying two powers with the same base, you add the exponents. For example, \(x^a \cdot x^b = x^{a+b}\).
• Power of a Power Rule: Discussed later in more detail.
• Zero Exponent Rule: Any non-zero number raised to the zero power is 1. For instance, \(x^0 = 1\), provided \(xeq0\).
• Negative Exponent Rule: A negative exponent indicates reciprocal, \(x^{-a} = \frac{1}{x^a}\).

Understanding and applying these rules makes simplifying complex expressions much easier.

Power of a Power Rule

The Power of a Power rule is a specific exponent rule used when an exponent is applied to another exponent. This rule states that you multiply the exponents together.

For example,

• If you have \( (x^a)^b \), it simplifies to \( x^{ab} \).
• Applied to numbers, if \( (2^3)^2 \), compute it as \( 2^{6} \) (because \( 2^{3 \times 2} = 2^{6} \)).

In the original exercise, this rule was applied to \((6x^3)^2\) and \((2x^2)^3\). By using the Power of a Power rule, \((6x^3)^2\) becomes \((6^2)x^6\) and \((2x^2)^3\) becomes \((2^3)x^6\). Applying this rule simplifies the way we handle expressions with nested exponents.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are algebraic expressions. Simplifying rational expressions involves similar processes to simplifying regular fractions.

Consider these steps when simplifying:

• Factor the numerator and denominator: This can sometimes help in canceling out terms.
• Cancel common terms: Simplify by canceling any terms that appear in both the numerator and the denominator.

With practice, rational expressions become easier to manage, and you'll find it straightforward to simplify expressions like \(\frac{36x^6}{8x^6}\) by first addressing any common factors or terms.

Greatest Common Divisor

The Greatest Common Divisor (GCD), or greatest common factor, is the largest number that can evenly divide two or more numbers. Finding the GCD is essential when reducing fractions to their simplest form.

For example, to simplify \(\frac{36}{8}\):

• Identify the GCD of 36 and 8: The divisors of 8 (1, 2, 4, 8) and 36 (1, 2, 3, 4, 6, 9, 12, 18, 36). The common divisors are 1, 2, and 4, with 4 being the greatest.
• Divide by the GCD: \(\frac{36 \div 4}{8 \div 4} = \frac{9}{2}\).

Dividing both the numerator and denominator by their GCD ensures the fraction is in its simplest form, increasing clarity and precision in algebraic operations.