Problem 50 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ \left(\frac{x y^{4}}{-3 z^{3}}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x^3 y^{12}}{-27 z^9} \).

Step 1: Distribute the Power to Each Component

Start by applying the power of a quotient rule, which says that \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \). This means that we need to distribute the power of 3 to both the numerator and the denominator: \( \left( \frac{x y^{4}}{-3 z^3} \right)^{3} = \frac{(x y^{4})^{3}}{(-3 z^{3})^{3}} \).

Step 2: Simplify the Numerator

In the numerator, you have \((x y^4)^3\). Apply the power rule which states that \( (a^m)^n = a^{m \times n} \). Thus, \( (x y^4)^3 = x^3 (y^4)^3 \). By using the power rule again for \( (y^4)^3 \), we get \( y^{4 \times 3} = y^{12} \). So, the numerator becomes \( x^3 y^{12} \).

Step 3: Simplify the Denominator

In the denominator, you have \((-3 z^3)^3\). Again, apply the power rule. First, \((-3)^3 = -27\), and then for \((z^3)^3\), it becomes \(z^{3 \times 3} = z^9\). Therefore, the denominator becomes \(-27z^9\).

Step 4: Write the Final Simplified Expression

Now that we have simplified both the numerator and the denominator, we can write the final simplified expression as: \( \frac{x^3 y^{12}}{-27 z^9} \). This expression cannot be simplified further using basic algebraic rules.

Key Concepts

Power of a Quotient Rule

The power of a quotient rule is a technique used in algebra to simplify expressions where both the numerator and the denominator of a fraction are raised to an exponent.
This rule follows the formula: \[\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}.\]This means when you have a fraction and an exponent outside the parenthesis, you can apply that exponent to both the top (numerator) and the bottom (denominator) separately.
In our exercise, the expression \(\left(\frac{x y^{4}}{-3 z^{3}}\right)^{3}\) utilizes this rule.
By distributing the power of 3 to both the numerator \((x y^{4})\) and the denominator \((-3 z^{3})\), you help to simplify the overall expression. Understanding this rule is essential because it allows you to systematically break down complex fractions with exponents into more manageable components.

Simplifying Expressions

Simplifying expressions is about making them as straightforward as possible. 
It involves processes that apply mathematical rules like combining like terms, eliminating unnecessary factors, and reducing expressions to the simplest form.
In the context of the exercise, once we've applied the power of a quotient rule to expand \[\left(\frac{x y^{4}}{-3 z^{3}}\right)^{3}\] into \[\frac{(x y^{4})^{3}}{(-3 z^{3})^{3}},\] we need to simplify further.Here, you'd break down both the numerator and denominator using methods like expanding powers over products. This way, it becomes easy to observe any opportunities to reduce or cancel terms.
After simplifying the numerator to \((x y^4)^3 = x^3 y^{12}\) and the denominator to \((-3)^3 z^9 = -27 z^9\), the expression simplifies clearly to \[\frac{x^3 y^{12}}{-27z^9}.\]The goal of simplifying is to achieve the simplest form, allowing for easier computations or further manipulation in solving equations.

Exponent Rules

Understanding exponent rules makes working with powers much easier and less error-prone.
These rules include

• The Product of Powers Rule: If you multiply like bases, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• The Power of a Power Rule: Raising a power to another power means you multiply the exponents: \( (a^m)^n = a^{m \times n} \).
• The Power of a Product Rule: Distribute the exponent to each factor in the product: \( (ab)^n = a^n b^n \).

Using these rules in our original problem utillized the power of a power rule, especially when applying it to the expressions such as \(y^{4}\) raised to the power of 3, resulting in \(y^{12}\).
Similarly, when dealing with the denominator \((-3)^3\), understanding these rules allows mathematicians to manage and manipulate algebraic expressions efficiently and accurately.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operators such as addition and multiplication.
These expressions can be quite complex, involving different rules, such as the exponent rules, to simplify them.

• Variables: These represent unknown values, like \(x\) and \(y\) in the expression.
• Constants: Fixed values that do not change, such as \(-3\) in our example.
• Coefficients: Numbers multiplying the variables, which can also be simplified using algebraic rules.
• Operators: Symbols that show the operation (add, subtract, multiply, divide) between terms.

The challenge usually involves organizing these elements according to algebraic rules. In our problem, understanding what each part of the expression represents helps in applying the correct rules.
For example, in \[\left(\frac{x y^{4}}{-3 z^{3}}\right)^{3},\] being able to identify the terms and factors involved is crucial before applying rules to simplify.
Algebraic expressions form the basis of algebra and learning to manipulate them provides powerful tools for solving equations and understanding more complex mathematical concepts.