Problem 74 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The simplified expression is \( z^{22} \).

Step 1: Write the Expression

The given expression is \( \left[z\left(z^{3} \cdot z\right)^{2} z^{2}\right]^{2} \). Our goal is to simplify this expression, starting from the innermost parentheses.

Step 2: Simplify Inside the Parentheses

Inside the parentheses, we have \( z^{3} \cdot z \). Using the law of exponents, \( a^{m} \cdot a^{n} = a^{m+n} \), we can simplify this to \( z^{3+1} = z^{4} \). Thus, the expression becomes \( \left[z\left(z^{4}\right)^{2} z^{2}\right]^{2} \).

Step 3: Apply the Exponent on \(z^{4}\)

Inside the brackets, \( \left(z^{4}\right)^{2} \) can be simplified using the power of a power rule: \( (a^{m})^{n} = a^{m \cdot n} \). This gives us \( z^{4 \cdot 2} = z^{8} \). The expression is now \( \left[z \, z^{8} z^{2}\right]^{2} \).

Step 4: Combine Terms Inside the Brackets

Now we combine the \( z \, z^{8} z^{2} \) terms using the law of exponents: \( a^{m} \cdot a^{n} = a^{m+n} \). So, \( z^{1 + 8 + 2} = z^{11} \). The expression simplifies to \( \left[z^{11}\right]^{2} \).

Step 5: Apply the Outside Exponent

Finally, apply the square to \( z^{11} \) using the power of a power rule: \( (a^{m})^{n} = a^{m \cdot n} \). So, \( (z^{11})^{2} = z^{22} \).

Step 6: State the Final Simplified Expression

The simplified form of the given expression is \( z^{22} \).

Key Concepts

Simplifying Algebraic Expressions

When we talk about simplifying algebraic expressions, we mean breaking down complex equations into a more manageable and understandable form. This involves using different rules and properties of numbers and variables to make expressions simpler and easier to work with.
Let's consider the expression \( \left[z\left(z^{3} \cdot z\right)^{2} z^{2}\right]^{2} \). Here, we apply order of operations and specific mathematical properties to simplify it step-by-step.

• Identify similar terms: Look for terms that can be combined, such as those that share the same variables and powers.
• Apply rules: Use properties like the distributive property or laws of exponents to combine and simplify terms.
• Reorganize terms: Change the order to group similar terms together, which may help in further simplification.

By systematically applying these steps, we reduce an algebraic expression to its simplest form, making further mathematical manipulation or evaluation more straightforward.

Laws of Exponents

Exponents, also known as powers or indices, are an essential part of algebra that indicate how many times a number is multiplied by itself. The laws of exponents are rules that simplify expressions involving powers and allow us to manipulate them easily.
Here are some key laws used when simplifying the given expression:

• Product of Powers Rule: \( a^{m} \cdot a^{n} = a^{m+n} \) helps to combine like bases by adding their exponents.
• Power of a Power Rule: \( (a^{m})^{n} = a^{m \cdot n} \) indicates you need to multiply the exponents.
• Power of a Product Rule: \((ab)^{n} = a^{n}b^{n}\) enables us to apply the exponent to each factor individually.

Using these laws, the expression \( z^{3} \cdot z \) simplifies to \( z^{4} \), and applying \((z^{4})^{2}\) becomes \( z^{8} \).
These laws transform complex exponential expressions into simpler forms, avoiding cumbersome calculations and mistakes.

Power of a Power Rule

The power of a power rule is a powerful exponentiation tool used in algebra for simplifying expressions where a power is raised to another power. This rule states: \((a^{m})^{n} = a^{m \cdot n}\). It tells us that when you raise a power to another power, you multiply the exponents.
Let's break it down using our example. In the expression \((z^{4})^{2}\), the power of a power rule allows simplifying it to \(z^{4 \cdot 2} = z^{8}\).
Here is a step-by-step approach:

• Identify the base and the powers involved: In \((z^{4})^{2}\), the base is \(z\), and the powers are \(4\) and \(2\).
• Multiply the exponents: Calculate \(4 \cdot 2\) to get \(8\).
• Write the simplified expression: Substitute the product back as the single power, resulting in \(z^{8}\).

This process condenses multi-layered exponential expressions, simplifying the way they are calculated and interpreted.
The main advantage is the avoidance of errors when dealing with large numbers or more complex algebraic functions.