Problem 68 - Brief 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: Distribute the Inner Exponent

First, distribute the exponent that is inside the parentheses. The expression inside is \( (z^3 \cdot z)^2 \). We start by applying the exponent to each factor:\[(z^3 \cdot z)^2 = (z^3)^2 \cdot (z)^2\]

Step 2: Apply Power Rule for Exponents

According to the power rule, \((x^a)^b = x^{a \cdot b}\). Apply this rule to each part:\[(z^3)^2 = z^{3 \times 2} = z^6\]\[(z)^2 = z^2\]Now replace in the equation:\[z (z^6 \cdot z^2) = z^{1+6+2} = z^9\]

Step 3: Simplify Inner Term

Combine the exponents by adding them:\[z \cdot z^6 \cdot z^2 \cdot z^2 = z^{1+6+2+2} = z^{11}\]

Step 4: Apply the Outer Exponent

The result from Step 3 is now the base for the outer exponent. Apply the exponent:\[(z^{11})^2 = z^{11 \cdot 2} = z^{22}\]

Step 5: Final Expression

The simplified form of the given expression is thus:\[z^{22}\]

Key Concepts

Power Rule for Exponents

The power rule for exponents is a fundamental concept in mathematics that provides a straightforward way to handle powers raised to other powers. When you have an expression like

• \((x^a)^b\)

The power rule tells us to multiply the exponents. It becomes:

• \(x^{a \,\times\, b}\)

This rule helps simplify complex expressions readily.
For example, in the original problem, the expression \((z^3)^2\) is simplified using this power rule to become:
\[z^{3 \,\times\, 2} = z^6\]This makes the rest of the calculation much easier, as it reduces multiple layers of exponents efficiently.
Always remember, when dealing with exponents inside the parentheses, use the power rule to multiply them.

Distribution Property in Algebra

The distribution property in algebra describes how we apply an operation across elements inside parentheses. When involving exponents, this often means distributing an outer (or second) exponent to each element inside the parentheses.
In the expression \((z^3 \cdot z)^2\), the exponent of 2 is distributed to both factors inside:

• The term \((z^3)^2\)
• The term \((z)^2\)

This distribution is like spreading a task evenly to make it simple. After distributing, each part is evaluated separately before combining.
The successful use of this property ensures each factor gets its respective treatment from the exponent, leading to a fully simplified single expression without errors. This step is crucial in maintaining the integrity of the equation as it breaks it down into more manageable parts.

Exponentiation in Algebra

Exponentiation is one of the essential algebraic operations where a number, known as the base, is raised to the power indicated by an exponent. When multiple bases are involved, like repeated use of the same base, simplify by applying exponent rules.
In algebra, if you encounter something like

• \(z^a \cdot z^b\)

you add the exponents together:

• \((z^{a+b})\)

Each base that is the same multiplies by simply adding up their exponents together, aided by the simplicity of exponent addition.
For example, combining steps in the exercise allowed the combining of multiple powers of the same base \(z\), leading to
\[z \cdot z^6 \cdot z^2 \cdot z^2 = z^{1+6+2+2} = z^{11}\] Taking each step methodically ensures you correctly interpret and simplify exponentiated terms, leading ultimately to a much simpler and concise result, like arriving at \(z^{22}\).