Problem 67 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Simplify inside the parentheses

Start by simplifying the expression inside the parentheses: \(z^{2}(z imes z^{2})^2 z\). First, simplify \(z \times z^{2}\) to get \(z^{1+2} = z^{3}\).

Step 2: Apply the power of a product

Rewrite the expression using the result from Step 1: \(z^{2}(z^{3})^{2}z\). Use the property of exponents that states \((a^m)^n = a^{mn}\) to simplify \((z^{3})^{2}\) to \(z^{3 \times 2} = z^{6}\).

Step 3: Combine exponents

Now, simplify \(z^{2}z^{6}z\) by adding the exponents since they have the same base. This yields \(z^{2+6+1} = z^{9}\).

Step 4: Raise the expression to the third power

The problem asks for the whole expression to be raised to the third power: \([z^{9}]^3\). Apply the power of a power rule: \((a^m)^n = a^{mn}\), which results in \(z^{9 \times 3} = z^{27}\).

Key Concepts

Exponent Rules

Exponents are a shorthand way to express repeated multiplication of the same number. Understanding and applying the rules for exponents can make complex problems much easier to solve.
In our exercise, we encounter expressions like \( z^2(z \cdot z^2)^2z \) that need simplification. The main rules that apply are:

• Product of Powers Rule: When multiplying two powers with the same base, you can add the exponents: \( a^m \times a^n = a^{m+n} \).
• Power of a Power Rule: When raising a power to another power, you multiply the exponents: \( (a^m)^n = a^{m\cdot n} \).
• Power of a Product Rule: When raising a product to a power, you apply the exponent to each factor inside the product: \( (ab)^n = a^n b^n \).

Applying these rules step-by-step helps to simplify complex expressions quickly. For example, in Step 1, applying the product of powers rule simplifies \( z \times z^2 = z^{1+2} = z^3 \).

Power of a Product

It’s important to understand how to deal with expressions where a product is raised to an exponent. This simplifies calculations significantly and helps solve problems like our exercise efficiently.
When given \((z^{3})^{2}\) in Step 2 of the solution, apply the power of a power rule wherein \( (a^m)^n = a^{m\cdot n} \).Using this rule, \( (z^{3})^{2} \) becomes \( z^{3\times2} = z^{6} \).

Therefore, raising \((z\cdot z^2)\) to the 2nd power simplifies to calculating each part of the product separately, which can be done by calculating \( z^3 \) (since \( z \times z^2 \) was simplified to \( z^3 \) in Step 1), raising it to \((z^{3})^{2}\) hence becoming \(z^{6}\).
This technique streamlines computations when handling polynomials and other algebraic expressions.

Combining Like Terms

Combining like terms involves simplifying expressions by adding or subtracting terms with the same variable raised to the same power. It's a fundamental skill in algebra that makes equations easier to handle.
In the example problem, once simplified to \( z^{2}z^{6}z \), we have terms that are all powers of the same base: \( z \).

Here, apply the rule that says multiply factors with the same base by adding their exponents: \( z^{2+6+1} = z^9 \).This combines all like terms into one, further simplifying the expression.
Finally, \([z^{9}]^3\) uses the power of a power rule again to yield \( z^{27} \).
Combining like terms is straightforward once you identify parts of expressions that share the same variables and ensure that only coefficients and exponents are calculated, pulling together scattered terms into a simple form.