Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Use positive rational exponents to rewrite each expression. Assume variables represent positive numbers. $$(\sqrt[5]{z})^{-3}$$

Step-by-Step Solution

Answer

The expression rewrites to \( \frac{1}{z^{\frac{3}{5}}} \).

Step 1: Understand the Radical Notation

The expression \( \sqrt[5]{z} \) represents the fifth root of \( z \). In general, the \( n \)-th root of a number can be written with an exponent of \( \frac{1}{n} \). So, \( \sqrt[5]{z} \) can be rewritten as \( z^{\frac{1}{5}} \).

Step 2: Apply the Negative Exponent

The given expression is \( (\sqrt[5]{z})^{-3} \). With our previous conversion, the expression \( \sqrt[5]{z} \) becomes \( z^{\frac{1}{5}} \). Now, apply the negative exponent: \( (z^{\frac{1}{5}})^{-3} \).

Step 3: Use the Property of Exponents

The property \((a^m)^n = a^{m \cdot n}\) allows us to multiply the exponents when raising a power to another power. So, \((z^{\frac{1}{5}})^{-3} = z^{\frac{1}{5} \cdot -3} \).

Step 4: Simplify the Exponent

Compute the product of the exponents: \( \frac{1}{5} \times -3 = -\frac{3}{5} \). Therefore, \( (z^{\frac{1}{5}})^{-3} = z^{-\frac{3}{5}} \).

Step 5: Convert to Positive Rational Exponent

To rewrite the expression with a positive rational exponent, use the negative exponent rule, \( a^{-m} = \frac{1}{a^m} \). This means \( z^{-\frac{3}{5}} = \frac{1}{z^{\frac{3}{5}}} \). This is the expression with positive rational exponents.

Key Concepts

Radical Notation

Radical notation may look complex, but it simply refers to roots, such as square roots or cube roots, that can be expressed in an alternate form. For example, the square root of a number, like 4, is written as \( \sqrt{4} \). However, mathematically, we can also express this as \( 4^{\frac{1}{2}} \). Here, the 2 in \( \sqrt{} \) (even if not written) means it's a square root, and is the denominator of the fraction in the exponent form.

In more general terms, the \( n \)-th root of \( x \) is symbolized as \( \sqrt[n]{x} \). This can be converted into the rational exponent form \( x^{\frac{1}{n}} \).

• Example: A fifth root, like in \( \sqrt[5]{z} \), becomes \( z^{\frac{1}{5}} \).
• This conversion allow us to use rules of exponents, making calculations easier.

This is why learning to switch between these two notations is so helpful.

Negative Exponents

Negative exponents might seem puzzling at first, but they're quite straightforward. The rule is: a negative exponent signifies a reciprocal. Hence, \( a^{-m} \) is equivalent to \( \frac{1}{a^m} \).

This tells us two things:

• The base is flipped to its reciprocal form.
• After flipping, the exponent becomes positive on the new base.

An example is \( (z^{\frac{1}{5}})^{-3} \). According to negative exponent rules, we convert it into \( \frac{1}{z^{\frac{3}{5}}} \).

The concept helps simplify expressions and makes computations cleaner, especially when transitioning into rational exponents.

Properties of Exponents

Understanding the properties of exponents is essential in working with them efficiently. These properties provide the tools needed to manipulate and solve expressions involving exponents.

Here are some key properties:

• Product of Powers: Adds the exponents of like bases: \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers: Subtracts exponents when dividing like bases: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: Multiplies the exponents when a power is raised to another power: \((a^m)^n = a^{m \cdot n}\).

These properties not only simplify calculations but also support converting complex expressions into a form that's easier to work with.

In the example with radicals and negative exponents, the property \((z^{\frac{1}{5}})^{-3}\) becomes \(z^{\frac{1}{5} \cdot -3}\), illustrating the power of a power property perfectly. This eventually simplified to \(z^{\-\frac{3}{5}}\), which was then converted to a positive exponent.