Problem 23 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

\( \sqrt[3]{z^{5}} = z^{\frac{5}{3}} \).

Step 1: Understand the Cube Root

The expression \( \sqrt[3]{z^{5}} \) denotes the cube root of \( z^{5} \). The cube root of a number \( x \) can be expressed as \( x^{\frac{1}{3}} \) when using rational exponents.

Step 2: Apply the Properties of Exponents

Using the property \( (a^{m})^{n} = a^{m \cdot n} \), rewrite \( \sqrt[3]{z^{5}} \) as an exponential expression. The cube root can be expressed as raising to the power of \( \frac{1}{3} \), so \( \sqrt[3]{z^{5}} = (z^{5})^{\frac{1}{3}} \).

Step 3: Simplify the Exponent

Apply the exponent rule by multiplying the exponents: \( (z^{5})^{\frac{1}{3}} = z^{5 \cdot \frac{1}{3}} = z^{\frac{5}{3}} \). This is the expression rewritten with positive rational exponents.

Key Concepts

Cube Root

A cube root is a special type of root, where you are trying to find a number that, when multiplied by itself three times, gives the original number. This is known as "cubing" a number. When you see the cube root symbol \( \sqrt[3]{x} \), it indicates the cube root of \( x \). For expressions involving exponents, it's often useful to express roots in terms of fractional exponents instead.

• The cube root of any number \( x \) can be written as \( x^{\frac{1}{3}} \).
• This transformation to a fractional exponent allows more flexibility in calculations, especially when paired with other exponents.

Understanding cube roots in terms of fractional exponents is particularly useful when simplifying expressions or when evaluating powers and roots collectively. Remember, cube roots help equate the number after being multiplied three times, while fractional exponents offer a neat and compact way to express the same concept.

Properties of Exponents

Exponents have several key properties that make calculations more manageable. Recognizing these properties makes it easier to manipulate and simplify expressions with exponents. One of the most useful properties is the power of a power rule.

• The power of a power rule states that when raising an exponent to another exponent, you multiply the exponents together: \( (a^m)^n = a^{m \cdot n} \).
• This property simplifies expressions considerably. For instance, by applying this property, \( (z^5)^{\frac{1}{3}} \) becomes \( z^{5 \cdot \frac{1}{3}} \).

Understanding these exponent properties allows us to transform complex or nested exponent expressions into simpler forms efficiently. This method of manipulations helps in both reformulating problems and solving them in algebraic operations.

Exponent Multiplication Rule

The exponent multiplication rule is a fundamental concept in working with expressions involving multiple exponents. This rule helps simplify the expressions when a base is raised to two different powers. Here's how it works:

• When you have a situation where two exponents are involved, such as \( a^m \) and another exponent \( n \) also affecting it, you multiply the exponents: \( a^{m \cdot n} \).
• This means if you start with \( (z^5)^{\frac{1}{3}} \), you expand it to \( z^{5 \times \frac{1}{3}} \).
• The rule ensures that exponent arithmetic becomes straightforward, cutting down multi-step algebra to more manageable pieces.

Using the exponent multiplication rule makes it straightforward to handle and simplify expressions that have layers of exponents. This not only saves time but also reduces the complexity of algebraic problems.