Problem 52 - Elementary Algebra

Question

Simplify \(\left(\frac{2 x^{0} y^{0} z^{3}}{z^{2}}\right)^{5}\).

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(\frac{2 x^{0} y^{0} z^{3}}{z^{2}}\right)^{5}\). Answer: \(32z^5\)

Step 1: Simplify the expression inside the parenthesis

First, recall that any non-zero number raised to the power of 0 is 1. Thus, we have: \(2 x^0 y^0 z^3 = 2 \cdot 1 \cdot 1 \cdot z^3 = 2 \cdot z^3\) Next, divide \(z^3\) by \(z^2\). Use the rule of exponents that \(a^{m} / a^{n} = a^{(m-n)}\), where \(a\) is a non-zero number and \(m\), \(n\) are real numbers. Then, we have: \(\frac{2 \cdot z^3}{z^2} = 2 \cdot z^{(3-2)} = 2z\) So, the expression inside the parenthesis is simplified as \(2z\).

Step 2: Apply the power of 5 to the simplified expression

Now, we have to raise the simplified expression \(2z\) to the power of 5. Use the power rule of exponents, \((ab)^n = a^n b^n\), where \(a\) and \(b\) are non-zero numbers and \(n\) is a real number. So, we get: \((2z)^5 = 2^5 \cdot z^5 = 32z^5\) The simplified expression is \(32z^5\).

Key Concepts

Simplification

When we talk about simplification in mathematics, we mean making an expression easier to work with or understand. Consider breaking down complex expressions into simpler parts. For example, in the original exercise, we start with the expression \[\left(\frac{2 x^{0} y^{0} z^{3}}{z^{2}}\right)^{5}\]and aim to simplify it.
We employ the zero exponent rule to recognize that terms like \( x^0 \) and \( y^0 \) equal 1. Thus, the expression inside the parentheses simplifies to \( 2z^3 \), further simplified by dividing by \( z^2 \). This results in \( 2z \).

• Identify components: Break down the parts of the expression.
• Apply rules: Use exponent and division rules.
• Combine results: Formulate a simpler expression post-adjustments.

Once the expression inside the parentheses is simplified, it becomes much easier to apply additional mathematical operations, such as raising the power.

Power Rules

Power rules govern the operations that deal with exponents in mathematical expressions. These rules provide a system to simplify handling powers. In our context, using power rules made simplifying \[(2z)^5 \]possible.
The general power rule states:

• \((ab)^n = a^n \cdot b^n\), which allows us to apply the exponent individually to factors in a product.
• \((a^m)^n = a^{m \cdot n}\), useful for nested exponents.

In the solution:- \( (2z)^5 \) simplifies to \( 2^5 \cdot z^5 \).
This translates to \( 32z^5 \) because \( 2^5 = 32 \). Understanding this power rule helps manage exponents when dealing with products and powers efficiently.

Division of Exponents

When dividing exponential expressions with the same base, a specific rule simplifies the process. This rule is particularly useful in managing expressions broken down into smaller partitions.
To divide exponents effectively:

• Use \( a^m / a^n = a^{(m-n)} \), where \(a\) is a non-zero base.

For example, consider the division: \[\frac{z^3}{z^2}\]This simplifies to \( z^{3-2} = z^1 = z \).As demonstrated, this rule also aligns with subtracting the exponents on identical bases when divided, producing fewer calculation steps.
Implementing this approach ensures expressions become manageable and eases the process of solving complex equations.

Zero Exponent Rule

The zero exponent rule is a fundamental rule in exponentiation. It states that any non-zero number raised to the power of zero is equal to 1.
This concept explains calculations involving zero exponents:

• \( x^0 = 1 \)
• \( y^0 = 1 \)

Considering the expression \( 2x^0y^0z^3 \), both \( x^0 \) and \( y^0 \) simplify to 1. Hence, the expression simplifies further to \( 2z^3 \).
In general, removing unnecessary components and simplifying the computational steps ensures clarity in calculations, leading to more straightforward problem-solving. Understanding the zero exponent rule allows you to clear easements in complex expressions, streamlining your solution path.