Problem 44 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ \left(2 z^{2}\right)^{-5} z^{10} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{32}\).

Step 1: Understand Negative Exponents

The expression \((2z^2)^{-5}\) has a negative exponent, which means we take the reciprocal of the base. So, \((2z^2)^{-5}\) is equal to \(\frac{1}{(2z^2)^5}\).

Step 2: Simplify the Power of a Product

Expand the power for the term \((2z^2)^5\). This means we raise both 2 and \(z^2\) to the 5th power: \(2^5(z^2)^5 = 32z^{10}\).

Step 3: Substitute Back into the Expression

Now substitute the simplified form back into the expression: \(\frac{1}{32z^{10}} \cdot z^{10}\).

Step 4: Simplify the Multiplication

Multiply \(\frac{1}{32z^{10}}\) by \(z^{10}\). This effectively cancels out \(z^{10}\) in the numerator and denominator, resulting in \(\frac{1}{32}\).

Step 5: Final Simplification

This leaves us with the simplified expression, \(\frac{1}{32}\), as there are no other terms to combine or simplify.

Key Concepts

Negative Exponents

Negative exponents might seem confusing at first, but they follow a simple rule. They tell us to take the reciprocal of the base while converting the exponent to positive. In other words, when you see a negative exponent, think about flipping the base. For instance, if you have a term like

• \(a^{-n}\), this is equivalent to \(\frac{1}{a^n}\).

This means that multiplying by an expression with a negative exponent is the same as dividing by the same expression raised to the positive exponent. This principle is crucial when we encounter expressions in algebra, as it helps us transform negative powers into more manageable terms. So, always remember: negative exponents are simply invitations to flip the base!

Simplification

Simplification in algebra involves reducing expressions to their simplest form so they can be more easily understood or worked with. This includes combining like terms, simplifying fractions, and reducing expressions as much as possible.
In the original problem, the simplification started by understanding and applying the properties of exponents. Once revised, allow for more straightforward operations:

• After applying the concept of negative exponents, we took the next steps to resolve the terms: changing \( (2z^2)^{-5} \) into \( \frac{1}{(2z^2)^5} \).
• Next, we combined terms, \( \frac{1}{(2z^2)^5} \times z^{10} \), focusing on canceling or reducing similar items in the numerator and denominator, especially common terms such as \( z^{10} \).

Recognizing how to factor and reduce expressions in algebra is a powerful tool. It transforms complex equations into more straightforward problems, making them easier to work with.

Power of a Product

When you raise a product to a power, each factor inside the parentheses must take on the exponentiation process. This is known as the power of a product rule. It tells us that:

• \((ab)^n = a^n \cdot b^n\)

This rule is applied during the simplification process in the problem, where \( (2z^2)^5 \) was expanded to \( 2^5 \cdot (z^2)^5 \). Let’s break this down:

• The factor 2 raised to the fifth power gives us \( 2^5 = 32 \).
• The factor \( z^2 \) raised to the fifth power follows the rule of multiplying exponents, resulting in \( (z^2)^5 = z^{10} \).

This distribution of the power is how we simplify complex exponential expressions. It allows large expressions to be broken down within the power structure, keeping calculations clear and manageable.