Problem 23 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(\frac{x^{2}\left(y^{3} z\right)^{3}}{\left(x^{2} y\right)^{2} z}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{y^{7} z^{2}}{x^{2}} \).

Step 1: Expand the Powers

Start by expanding the power in the numerator: \( (y^3 z)^3 = y^{3 \times 3} z^{3} = y^9 z^3 \).This makes the expression: \( \frac{x^{2} y^{9} z^{3}}{(x^{2} y)^{2} z} \).

Step 2: Expand the Denominator

Now, expand the power in the denominator: \( (x^{2} y)^{2} = x^{2 \times 2} y^{2} = x^4 y^2 \).This gives the expression: \( \frac{x^{2} y^{9} z^{3}}{x^{4} y^{2} z} \).

Step 3: Simplify the Expression

Reduce the fraction by canceling out similar base terms:- For \( x \): \( x^{2} / x^{4} = x^{2-4} = x^{-2} \), which is \( \frac{1}{x^{2}} \).- For \( y \): \( y^{9} / y^{2} = y^{9-2} = y^{7} \).- For \( z \): \( z^{3} / z = z^{3-1} = z^{2} \).Thus, the simplified expression is \( \frac{y^{7} z^{2}}{x^{2}} \).

Step 4: Write the Final Simplified Result

Combine the simplified terms: The expression now becomes\[ \frac{y^{7} z^{2}}{x^{2}} \].This is the simplest form of the given algebraic expression.

Key Concepts

Exponent Rules

Exponent rules are essential for simplifying expressions with powers. These rules help us handle terms with the same base raised to different powers more efficiently. Here are some key rules to keep in mind:

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

Understanding these rules makes it easier to manipulate expressions and reach the simplest form. As seen in the solution, applying the product and quotient rules efficiently simplifies the original expression.

Rational Expressions

A rational expression is essentially a fraction where the numerator and the denominator are polynomials. Simplifying these can often involve reducing them by factoring or using exponent rules. Here are some steps to simplify rational expressions:

• Factor Both Numerator and Denominator: Look for common factors and reduce them.
• Apply Exponent Rules: Use rules like the product and quotient of powers to simplify terms.
• Simplify Coefficients: Reduce numerical coefficients if possible.

The given exercise involves simplifying multiple terms with different variables and exponents. By systematically applying rules to each part of the expression, we can reduce it to its simplest form, much like basic numeric fractions.

Power of a Power Rule

The power of a power rule is a very efficient exponent rule that helps to simplify expressions where one power is raised to another power. The key to applying this rule is:

• Multiply the exponents: \((a^m)^n = a^{m \times n}\).

This rule simplifies expressions quickly by consolidating multiple exponentiations into a single operation. For example, in the step-by-step solution, this rule was applied to both the numerator and the denominator to initially expand and then reduce the given expression. Recognizing opportunities to use this rule can help make the process of simplification faster and more straightforward.