Problem 38 - Precalculus: Functions and Graphs

Question

Simplify. $$\left(25 z^{4}\right)^{-3 / 2}$$

Step-by-Step Solution

Answer

\(\frac{1}{125z^6}\)

Step 1: Apply the Power of a Power Rule

The power of a power rule states that when you have a power raised to another power, you multiply the exponents. In this case, distribute the exponent \(-\frac{3}{2}\) to both the base \(25\) and the base \(z^4\):\[(25^{1} \cdot z^{4})^{-\frac{3}{2}} = 25^{-\frac{3}{2}} \cdot (z^4)^{-\frac{3}{2}}\].

Step 2: Simplify 25 to the Power of -3/2

The expression \(25^{-\frac{3}{2}}\) can be simplified by first finding the square root of 25, and then raising it to the power of \(-3\). Since \(\sqrt{25} = 5\), we have:\[25^{-\frac{3}{2}} = (5^2)^{-\frac{3}{2}} = 5^{-3} = \frac{1}{5^3} = \frac{1}{125}\].

Step 3: Simplify \(z^{4}\) to the Power of -3/2

Similarly, apply the exponent \(-\frac{3}{2}\) to \(z^4\), which can be rewritten as:\[(z^4)^{-\frac{3}{2}} = z^{4 \cdot (-\frac{3}{2})} = z^{-6}\].

Step 4: Combine Results

Now, combine the results from Step 2 and Step 3:\[\frac{1}{125} \cdot z^{-6} = \frac{1}{125z^6}\].

Key Concepts

Power of a Power Rule

The "Power of a Power Rule" is a fundamental tool in the world of exponents. It simplifies expressions significantly by eliminating the need for multiple calculations. When you see a power being raised to another power, like \((x^m)^n\), this rule tells us that we can multiply these exponents together. Instead of performing two separate power operations, you simply perform one multiplication with the exponents.
Example: If you have an expression \((a^3)^4\), it can be simplified as \(a^{3 \times 4} = a^{12}\).
In the context of our problem, we applied this rule to distribute the outer exponent \(-\frac{3}{2}\) across the bases inside the parentheses. So, \( (25^{1} \cdot z^{4})^{-\frac{3}{2}} \) became \( 25^{-\frac{3}{2}} \cdot (z^4)^{-\frac{3}{2}} \). This simplicity is the strength of the power of a power rule.

Negative Exponents

Negative exponents can seem tricky at first, but they are a way to express division in exponent terms. A negative exponent indicates that the base should be on the opposite side of a fraction line. Specifically, \(a^{-n} \) is equal to \(\frac{1}{a^n}\). This essentially "flips" the base to the denominator of a fraction.
For example, if \(b^{-2}\) is given, it is equivalent to saying \(\frac{1}{b^2}\). This is particularly helpful when simplifying equations, as moving bases with negative exponents can make calculations more manageable.
In our exercise, we applied a negative exponent to both 25 and \(z^4\). For \(25^{-\frac{3}{2}}\), it helped to transform the expression to a fraction by moving it from its position to the denominator. Similarly, \(z^{-6}\) became \(\frac{1}{z^6}\). Through simplifying like this, we make the expression easier to handle.

Simplifying Exponents

Simplifying exponents involves making expressions with powers easier and neater. It includes using rules like those for multiplying, dividing, and taking powers of exponents. Plus, it involves transforming negative exponents into fractions.
Sometimes you'll start by using the power of a power rule or converting negative exponents into fractions, as we did earlier. Then, combine the results in the simplest form possible, maintaining proper mathematical rules.
In our exercise, after employing the power of a power rule and understanding negative exponents, we're left with each part in a simple fraction form. The last task was to combine the processed components. That meant writing them under a single division, which simplified \(\frac{1}{125} \cdot z^{-6}\) into \(\frac{1}{125z^6}\). The result is much easier to interpret or apply in further calculations.