Problem 73 - Applied Calculus

Question

Simplify. $$ \left[z^{2}\left(z \cdot z^{2}\right)^{2} z\right]^{3} $$

Step-by-Step Solution

Answer

The simplified form is \( z^{27} \).

Step 1: Distribute the Inner Exponents

Let's start by working inside the innermost parentheses: \[ z imes z^2 = z^{1+2} = z^3 \]So the expression \[ z^2 imes (z imes z^2)^2 imes z \]becomes: \[ z^2 imes (z^3)^2 imes z \]

Step 2: Simplify the Power inside Parentheses

Now we have to compute \[ (z^3)^2 \]To do this, multiply the exponents: \[ z^{3 imes 2} = z^6 \]

Step 3: Combine the Remaining Exponents

Replace back into the expression: \[ z^2 imes z^6 imes z \]Combine the exponents: \[ z^{2+6+1} = z^9 \]

Step 4: Compute the Outer Exponentiation

The expression \[ [ z^9 ]^3 \]involves raising a power to a power. Multiply the exponents: \[ z^{9 imes 3} = z^{27} \]

Step 5: Final Answer

So, the simplified form of the given expression is: \[ z^{27} \]

Key Concepts

Exponentiation

Exponentiation is a fundamental concept in algebra that involves raising a number or variable to a power. In mathematics, if you see an expression like \( z^n \), it means multiplying the base \( z \) by itself \( n \) times. This concept is crucial when dealing with algebraic expressions and equations.
For example, \( z^2 \) means \( z \times z \), and \( z^3 \) represents \( z \times z \times z \). These expressions are often encountered in various scenarios, especially when simplifying complex equations. Exponentiation helps in expressing large numbers in a compact form, making calculations more manageable.
In the given exercise, exponentiation is used multiple times to handle the powers as the expression is simplified. Understanding exponentiation helps to make the transition from a complex expression to a simpler one.

Power Laws

Power laws are a set of rules that govern how to handle expressions involving exponents. These laws are essential for simplifying expressions and solving algebraic equations. Here are the key power laws:

• Product of Powers Rule: \( a^m \times a^n = a^{m+n} \) - When multiplying like bases, add the exponents.
• Power of a Power Rule: \( (a^m)^n = a^{m \times n} \) - When raising a power to another power, multiply the exponents.
• Power of a Product Rule: \( (ab)^n = a^n \times b^n \) - Distribute the exponent to both factors inside the parentheses.

In the solution provided, these laws are applied step by step:
First, the product of powers law is used to combine \( z \) and \( z^2 \) into \( z^3 \). Next, the power of a power rule simplifies \( (z^3)^2 \) to \( z^6 \). Finally, combining all exponents in one expression reinforces the simplicity provided by knowing these power laws.

Simplifying Expressions

Simplifying expressions is the process of transforming a complex expression into a simpler, more understandable form. The primary goal is to make calculations easier by reducing the number of terms and condensing the expression.
In our problem, we start with a rather intricate expression and through different steps, using power laws and multiplication, we simplify it down:

• First, deal with the inner part of the expression to remove extra parentheses.
• Next, apply exponent rules step-by-step to combine like bases.
• Finally, tackle any remaining powers to see the reduced form.

During the solution process, each maneuver with exponents aids in breaking the expression apart. By simplifying effectively, one can avoid miscalculations and make the tedious tasks more straightforward. The final answer, \( z^{27} \), is achieved by incrementally working through the layers of exponents to achieve the simplest form.