Problem 28 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

\( \frac{1}{z^{3/5}} \)

Step 1: Convert Radical to Exponential Form

First, recognize that the expression \( \sqrt[5]{z} \) is equivalent to \( z^{1/5} \). In mathematical terms, a radical can be expressed as a rational exponent: \[ \sqrt[n]{a} = a^{1/n} \] Therefore, \( \sqrt[5]{z} = z^{1/5} \).

Step 2: Apply the Negative Exponent Rule

Next, apply the rule of exponents for a negative exponent: \((a^m)^{-n} = a^{-m \cdot n} \) In our case, this translates to: \[(z^{1/5})^{-3} = z^{-(1/5) \cdot 3}\] This simplifies to \( z^{-3/5} \).

Step 3: Express as Positive Rational Exponent

The expression \( z^{-3/5} \) can also be rewritten using positive exponents, since the directions ask for positive rational exponents. To convert a negative exponent into a positive one, use the property:\[ a^{-b} = \frac{1}{a^b} \]Thus, \( z^{-3/5} \) becomes \( \frac{1}{z^{3/5}} \).

Key Concepts

Radical Expressions

Radical expressions are a way to represent the roots of numbers. When you see a radical sign like \( \sqrt[n]{a} \), it means you're looking for the number that, when raised to the power of \( n \), gives \( a \). For example, \( \sqrt{9} = 3 \) because \( 3^2 = 9 \). Radicals can be represented using rational exponents, which are much easier to work with, especially when dealing with algebraic expressions. The transformation from a radical to an exponent follows this rule:

• \( \sqrt[n]{a} = a^{1/n} \)

This is helpful as it simplifies operations like multiplication and division with similar terms. In our exercise, converting \( \sqrt[5]{z} \) into \( z^{1/5} \) helps to streamline further calculations. Remember, this process hinges on expressing roots as fractional exponents.

Negative Exponents

Negative exponents can initially seem confusing, but they actually have a simple explanation. A negative exponent indicates that the base should be taken as a reciprocal. For instance, \( a^{-1} \) is the same as \( \frac{1}{a} \). This means instead of multiplying by the base, you divide by it, essentially "flipping" the fraction.For any base \( a \) raised to a negative exponent \( n \):

• \( a^{-n} = \frac{1}{a^n} \)

Using this rule can help convert expressions with negative exponents into a form that’s often easier to interpret and calculate. This is why in the exercise, \( z^{-3/5} \) becomes \( \frac{1}{z^{3/5}} \). By transforming negative exponents this way, you can manipulate expressions to suit the requirements of a problem like expressing all exponents positively.

Exponentiation Rules

Exponentiation rules streamline the process of working with powers, which is extremely useful in simplifying complex algebraic expressions. There are some key rules to remember that make working with exponents much easier:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \)
• Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)

These rules apply to both positive and negative exponents. For instance, when faced with an expression such as \((z^{1/5})^{-3}\), the power of a power rule transforms this into \( z^{-(1/5)\cdot 3} \), simplifying to \( z^{-3/5} \). The initial confusion with exponents can ease away once you become familiar with these key properties.