Problem 28 - Algebra and Trigonometry

Question

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: Apply Power of a Power Property

When an expression with a power is raised to another power, multiply the exponents. So, we have \((2z^2)^{-5} = 2^{-5} \cdot (z^2)^{-5}\). This will simplify to \(2^{-5}z^{-10}\).

Step 2: Simplify using Negative Exponents

Recall that a negative exponent means taking the reciprocal of the base. Thus, \(2^{-5}\) becomes \(\frac{1}{2^5}\) and \(z^{-10}\) becomes \(\frac{1}{z^{10}}\).

Step 3: Express as a Single Fraction

Combine the terms from Step 2 to form a single fraction:\[ \frac{1}{2^5} \cdot \frac{1}{z^{10}} = \frac{1}{2^5 z^{10}}.\]

Step 4: Combine with Remaining Terms

Multiply the expression from Step 3 by \(z^{10}\) (the term given in the expression):\[ \frac{1}{2^5 z^{10}} \cdot z^{10}.\] Here, \(z^{10} \cdot \frac{1}{z^{10}} = 1\).

Step 5: Simplify Final Expression

The expression simplifies to \(\frac{1}{2^5}\) since the \(z^{10}\) terms cancel out. Calculating \(2^5\) gives \(32\). Thus, \(\frac{1}{2^5} = \frac{1}{32}\).

Key Concepts

Power of a Power Property

The power of a power property is a fundamental rule in exponentiation. When you have a base raised to an exponent, and then that entire term is raised to another exponent, you simply multiply the exponents. For example, in validating the expression \((2z^2)^{-5}\), the property is applied by multiplying the powers: \((z^2)^{-5}\) becomes \(z^{2 imes (-5)}\), resulting in \(z^{-10}\). Likewise, \(2^{-5}\) stems from the same rule. This property not only helps simplify complex expressions but also makes it easier to handle calculations involving multiple layers of exponents. Remember, it’s about multiplying, not adding the exponents.

Negative Exponents

Negative exponents might seem confusing at first, but they simplify expressions significantly. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. In other words, \(x^{-n} = \frac{1}{x^n}\). When simplifying the expression \(2^{-5}\), it becomes \(\frac{1}{2^5}\). Similarly, \(z^{-10}\) is converted to \(\frac{1}{z^{10}}\).

• To shift a term from the numerator to the denominator, change its exponent from negative to positive.
• If the term is in the denominator, a negative exponent shifts it to the numerator.

This rule ensures that expressions are easier to manipulate and solve.

Simplifying Expressions

Simplifying expressions involves reducing them into the simplest form possible. This makes equations easier to understand and solve. In the given expression \(\left(2 z^{2}\right)^{-5} z^{10}\), we start by handling the exponents with the power of a power property and negative exponents.

• Combine terms like \(2^{-5}\) and \(z^{-10}\) by using the rule of multiplying exponents and changing negative exponents to fractions.
• Once in a fractional form, integrate any remaining terms, like \(z^{10}\), by multiplying or cancelling.

Eventually, you end up with \(\frac{1}{2^5}\), which is simplified further by calculating the power as \(2^5 = 32\), resulting in \(\frac{1}{32}\). Maintaining clarity through each step streamlines solving complex algebraic expressions.