Problem 116 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \frac{c^{2 / 3} c^{2 / 3}}{c^{1 / 3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(c\).

Step 1: Simplify the Numerator

First, we simplify the numerator, which is a product of powers of the same base. Using the property of exponents \(a^m \cdot a^n = a^{m+n}\), the numerator becomes:\[c^{2/3} \cdot c^{2/3} = c^{2/3 + 2/3} = c^{4/3}\]

Step 2: Apply the Division of Exponents Rule

Next, apply the division rule for exponents \(\frac{a^m}{a^n} = a^{m-n}\) to the expression \(\frac{c^{4/3}}{c^{1/3}}\). This simplifies as follows:\[c^{4/3 - 1/3} = c^{3/3} = c^1\]

Step 3: Simplify the Power

Since \(c^1\) is simply \(c\), we can express the simplified form as:\[c\]

Key Concepts

Exponents

Exponents are a way of expressing repeated multiplication in a compact form. When we have an exponent, it tells us how many times to multiply the base by itself. For example, in the expression \(c^3\), \(c\) is the base, and the exponent \(3\) means that \(c\) is multiplied by itself three times: \(c \cdot c \cdot c\).
Exponents can take on fractional values as well. This is particularly useful when dealing with roots and radical expressions. For example, \(c^{1/3}\) represents the cube root of \(c\), and is equivalent to \(\sqrt[3]{c}\).
The key to mastering exponents is understanding their behavior with different operations, such as multiplication, division, and raising powers to powers.

Properties of Exponents

Understanding the properties of exponents helps simplify complex expressions. One of the most useful properties is the product of powers rule: whenever you multiply numbers with the same base, you can add the exponents. For example, \(a^m \cdot a^n = a^{m+n}\). This rule is handy in our original exercise, where we combined \(c^{2/3}\) and \(c^{2/3}\) to become \(c^{4/3}\).
Another important property is the quotient of powers rule, which states that when dividing numbers with the same base, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\). In the exercise, this rule allowed us to transform \(\frac{c^{4/3}}{c^{1/3}}\) into \(c^{3/3}\).
Finally, remember that any number to the power of 1 is itself, shown when we simplified \(c^1\) to \(c\). These small but significant steps make handling exponents much easier and more intuitive.

Positive Real Numbers

In mathematics, it is essential to understand what positive real numbers entail, especially while dealing with expressions involving exponents. Positive real numbers are any numbers greater than zero and can be expressed on the number line. Crucially, when the exercise states that all variables represent positive real numbers, it ensures certain operations are always valid.
For instance, taking roots of negative numbers would lead to complex numbers, which aren't usually part of simplified expressions at this level. Positive real numbers guarantee that every root operation results in a straightforward real number.
In our original exercise, using positive real numbers assures us that when we simplify the expression \(c^{1/3}\), \(c\) is inherently a positive value, making the process seamless and free of complex numbers. This clarity allows for more focus on applying the rules of exponents efficiently.