Question

Challenge Problems. $$\left(\frac{z^{0} z^{2 n-2}}{z^{n}}\right)^{3}$$

Short answer

\(z^{3n - 6}\)

Step-by-step solution

Simplify Exponents Inside the Parenthesis

Use the property of exponents that states: for any non-zero number a, and integers m and n, the product of a raised to the power of m and to the power of n is equal to a raised to the power of (m + n). For the term inside the parentheses, apply the property to simplify it as follows: \( z^{0} z^{2n - 2} = z^{(0 + 2n - 2)} = z^{2n - 2} \).

Apply Quotients of Powers Property

Use the quotient of powers property which states that for any non-zero number a and integers m and n, \(a^{m-n} = \frac{a^m}{a^n}\). Substitute \(z^{2n-2}\) and \(z^n\) into this property to find the simplified form of the fraction: \( \frac{z^{2n - 2}}{z^n} = z^{(2n - 2) - n} = z^{n - 2} \).

Raise the Simplified Term to the Power of 3

Now apply the power rule for exponents which states that \((a^m)^n = a^{mn}\). Raise the term \(z^{n - 2}\) to the power of 3 to find its cube: \( (z^{n - 2})^3 = z^{3(n - 2)} = z^{3n - 6} \).

Key concepts

Simplifying Exponents

Understanding how to simplify exponents is critical when working with algebraic expressions. It involves the process of reducing expressions with powers to their simplest form using various exponent rules.

For instance, in the given exercise, the first step is to simplify the exponent inside the parenthesis. By using the exponent product rule, which dictates that when you multiply powers with the same base, you add the exponents, we can combine terms with the base of 'z'.

Let's consider an example similar to the exercise:

1. Starting with terms like \( z^{0}z^{x} \), recall that any non-zero number raised to the power of 0 is 1. Hence, \( z^{0} \) simplifies to 1, and multiplying 1 by any term does not change its value.
2. Next, combining \( z^{0}z^{x} \) yields \( z^{x} \) because \( z^{0} \) acts as the multiplicative identity.

Following this approach, we can clear any confusion around simplifying exponents and ensure a solid understanding of one of the fundamental exponent properties.

Quotient of Powers Property

The quotient of powers property is a powerful tool when dividing expressions with the same base raised to different exponents. This property states that for any non-zero base 'a' and integers 'm' and 'n', the expression \(\frac{a^m}{a^n}\) simplifies to \(a^{m-n}\).

Applying this rule can significantly streamline complex expressions. Here's how it works:

• Identify the base that appears in both the numerator and denominator of the fraction.
• Subtract the exponent in the denominator from the one in the numerator.
• Write the result as the base raised to the new exponent.

Looking at the given exercise, once the exponent in the numerator is simplified, we apply this property to divide \(z^{2n-2}\) by \(z^n\), yielding \(z^{(2n-2)-n}\). Simplifying further, we arrive at \(z^{n-2}\), demonstrating a clear application of the quotient of powers property.

Power Rule for Exponents

The power rule for exponents provides a method for dealing with expressions where an exponent is raised to another exponent, like \((a^m)^n\). According to this rule, such an expression simplifies to \(a^{mn}\), which means you multiply the exponents together.

When applying this rule in sequences of calculations, it helps to follow these steps:

• Start by identifying the base and the two exponent levels.
• Multiply the exponents to consolidate them into a single exponent.
• Express the result as the base raised to this new exponent.

By doing so, you can confidently raise a power to another power, as seen in the third step of our exercise. From the simplified form \(z^{n-2}\), we use the power rule to find the cube of this expression, resulting in \(z^{3(n-2)}\), which further simplifies to \(z^{3n-6}\). This illustrates the elegance and efficiency of the power rule for exponents in action.