Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(25 z^{4}\right)^{-3 / 2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{125z^6}\).

Step 1: Understand Exponent Notation

First, notice that the expression is written using exponent notation with a negative fractional exponent, \(-\frac{3}{2}\). This means we have both a negative exponent and a root involved.

Step 2: Apply Negative Exponent Rule

The negative exponent \(-\frac{3}{2}\) indicates that we should take the reciprocal of the base inside the parentheses: \(\left(25z^{4}\right)^{-\frac{3}{2}} = \frac{1}{\left(25z^{4}\right)^{\frac{3}{2}}}\).

Step 3: Separate Base and Use Exponent

Break down \(\left(25z^{4}\right)^{\frac{3}{2}}\) into separate bases: \(25^{\frac{3}{2}}\) and \((z^{4})^{\frac{3}{2}}\).

Step 4: Simplify the Base 25

To simplify \(25^{\frac{3}{2}}\), recognize that \(\frac{3}{2}\) suggests performing a square root followed by cubing: \((\sqrt{25})^3 = 5^3 = 125\).

Step 5: Simplify the Power of z

Utilize the power of a power property \( (z^{4})^{\frac{3}{2}} = z^{\frac{4 \times 3}{2}} = z^{6} \).

Step 6: Combine Simplified Results

Substitute the simplified terms back together: \(\frac{1}{125z^6}\).

Step 7: Write the Final Simplified Answer

The fully simplified form of the expression is \(\frac{1}{125z^6}\).

Key Concepts

Negative Exponents

When dealing with exponents, a negative exponent may seem a bit intimidating. But don't worry, it's quite simple once you understand the basic rule. A negative exponent indicates that you should take the reciprocal of the base. This means that instead of multiplying the base by itself, we divide 1 by the base raised to the corresponding positive exponent.

Here's an easy to follow rule:

• If you have an expression like \(b^{-n}\), it becomes \(\frac{1}{b^n}\).

In the context of our exercise,

• \((25z^4)^{-\frac{3}{2}}\) transforms to \(\frac{1}{(25z^4)^{\frac{3}{2}}}\).

By using this rule, we transform the expression and make simplifying the fraction more straightforward.

Fractional Exponents

Fractional exponents might sound mysterious at first, but they are just another way of representing roots and powers in a single expression. When you see a fractional exponent, think of it as a combination of two operations: a power and a root.

• The numerator of the fraction tells you the power.
• The denominator tells you the root.

For example, in an expression like \(a^{\frac{m}{n}}\):

• The base \(a\) is raised to the power \(m\).
• Then, you take the \(n\)th root.

In our problem, \( (25z^4)^{\frac{3}{2}} \) involves taking the square root (since the denominator is 2), followed by raising to the third power. This allows us to simplify the expression step-by-step by addressing each part individually.

Power of a Power

The power of a power property is a handy rule when simplifying expressions with exponents. This rule states that when an exponent is raised to another exponent, you multiply the exponents together.

Essentially,

• If you have \((a^m)^n\), it simplifies to \(a^{m \times n}\).

In our exercise

• \((z^4)^{\frac{3}{2}}\) translates to \(z^{4 \times \frac{3}{2}} = z^6\).

This shows how the seemingly complex exponents become simpler by using the power of a power rule step by step. It breaks down the problem into manageable pieces, allowing clarity and an easier path to the solution.

Reciprocal

A reciprocal is a number which, when multiplied by the original number, results in the product of one. In mathematical terms, the reciprocal of a number \(x\) is \(\frac{1}{x}\). The concept of reciprocals is crucial when working with negative exponents, as it allows us to switch from negative to positive exponents by transforming the structure of the expression.

The logical reason behind using the reciprocal is that a negative exponent, as explained earlier, indicates division rather than multiplication. Thus,

• For \( (b^{-n})\),
  • it becomes \( \frac{1}{b^n} \).

Applying this to the given problem, turning

• \((25z^4)^{-\frac{3}{2}}\) into \(\frac{1}{(25z^4)^{\frac{3}{2}}}\).

The reciprocal approach simplifies negative exponentiation by reorganizing expressions in a more digestible format.