Problem 77 - Applied Calculus

Question

Simplify. $$ \left(3 x^{2} y^{5} z\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(27x^6y^{15}z^3\).

Step 1: Apply the Power to Each Factor Individually

When a product inside the parentheses is raised to a power, apply that power to each factor inside the parentheses. The expression is \((3 x^2 y^5 z)^3\). So, each element inside will be raised to the 3rd power: \(3^3\), \((x^2)^3\), \((y^5)^3\), and \(z^3\).

Step 2: Calculate the Coefficient Raised to the Power

Calculate \(3^3\). This is simply \(3 \times 3 \times 3 = 27\).

Step 3: Apply the Power to the Variables

Raise each variable to the power of 3. For \((x^2)^3\), use the power of a power rule, \((a^m)^n = a^{m \cdot n}\). Thus, \((x^2)^3 = x^{2 \cdot 3} = x^6\). Similarly, \((y^5)^3 = y^{5 \cdot 3} = y^{15}\) and \(z^3 = z^3\).

Step 4: Combine All Parts

Combine all elements calculated: \(27 x^6 y^{15} z^3\). This gives the final simplified form of the expression.

Key Concepts

Exponentiation Rules

Exponentiation rules are essential when working with powers and exponents in algebra. These rules help simplify expressions that involve exponential terms. There are a few principles you should always remember.

• Product of Powers Rule: When multiplying with the same base, add the exponents. For example, if you have \(a^m \times a^n\), this simplifies to \( a^{m+n} \).
• Power of a Power Rule: When raising an exponent to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
• Power of a Product Rule: When raising a product to a power, apply the exponent to each factor separately: \((ab)^n = a^n \times b^n\).

These basic rules apply to any real numbers and allow for effective simplification of even complex algebraic expressions. Grasping these concepts will make solving exercises involving powers much more straightforward.

Power of a Product Rule

The Power of a Product Rule is a technique used to handle expressions where multiple different factors are raised to a power. This rule can often simplify products that involve exponents. Suppose you have an expression such as \((ab)^n\).

This is handled by applying the exponent \(n\) to each factor inside the bracket individually, resulting in \(a^n \times b^n\). This method ensures each component of the product is elevated to the specified power.

For example, consider the expression \((3x^2y^5z)^3\):

• Apply the power 3 to the coefficient: \(3^3 = 27\).
• Apply the power to \(x^2\), giving \((x^2)^3 = x^{6}\).
• Apply the power to \(y^5\), giving \((y^5)^3 = y^{15}\).
• Finally, address \(z\), giving \(z^3\).

By using the Power of a Product Rule, you maintain clarity and simplicity while analyzing and simplifying expressions with multiple factors.

Simplifying Expressions

Simplifying expressions involving exponents is a key skill in algebra. It involves reducing expressions to their simplest form, making them easier to work with. Here are the foundational steps:

• Identify and group similar terms and variables within the expression.
• Apply relevant exponentiation rules, such as the Power of a Product Rule, to manage powers efficiently.
• Calculate constants separately, ensuring clarity and correctness.
• Combine the terms into a final, simplified expression.

Consider the expression \((3x^2y^5z)^3\). Each variable and number is addressed using the applicable rules:

• First, calculate \(3^3\), resulting in 27.
• Using the Power Rule, combine the exponents of the variable \(x\) to get \(x^6\).
• Do the same for \(y\), resulting in \(y^{15}\).
• Apply the power to \(z\), yielding \(z^3\).

The final expression, then, is neatly organized as \(27x^6 y^{15} z^3\). Learning and using these steps empowers you to simplify expressions effortlessly, ensuring accuracy in more complex algebraic problems.