Problem 24 - Algebra 2

Question

Write each expression in exponential form. $$\sqrt[4]{c^{2}}$$

Step-by-Step Solution

Answer

The exponential form of the expression \(\sqrt[4]{c^{2}}\) is \(c^{1/2}\)

Step 1: Understanding the problem

The expression is a fourth root of the square of a number \(c\), \(\sqrt[4]{c^{2}}\). This can also be written using the reciprocal power rule. The reciprocal power rule says that any root, such as a fourth root, can be expressed as a fraction in the exponent, 1/4 in this case.

Step 2: Applying the power rule

The expression \(\sqrt[4]{c^{2}}\) is equivalent to \(c^{2/4}\) using the power rule.

Step 3: Simplifying the fraction

To simplify, we divide the exponent \(2/4\) to get \(c^{1/2}\).

Key Concepts

Reciprocal Power Rule

In mathematics, roots can often be expressed using fractional exponents. This technique leverages the reciprocal power rule. When dealing with roots, such as square roots, cube roots, or in this case, the fourth root, the exponent is a fraction. The denominator of this fraction represents the root. For example, the fourth root of a number can be rewritten with an exponent of \(1/4\). Implementing this rule allows for simpler expressions and calculations, especially when combined with other mathematical rules and properties. So, applying the reciprocal power rule to \(\sqrt[4]{c^2}\), we convert it to \(c^{2/4}\). Here, the denominator \(4\) directly corresponds to the fourth root mentioned. This transformation is crucial for further simplification or calculation of the expression.

Fractional Exponents

Fractional exponents are a compact way to represent roots and powers simultaneously. In the expression \(c^{2/4}\), the fraction \(\frac{2}{4}\) is the exponent where the numerator \(2\) indicates the power to which the base \(c\) is raised, and the denominator \(4\) signifies the root being taken. This notation is especially useful in solving equations and simplifying expressions. By understanding and applying fractional exponents, we can easily transition between root and exponential forms of an expression, allowing for straightforward manipulation and simplification.For example, when you encounter a fractional exponent like \(c^{1/2}\), you immediately recognize it as the square root of \(c\). In this way, fractional exponents serve as a bridge between different forms of expressing numbers.

Simplifying Exponents

Simplifying exponents is a fundamental skill in algebra that makes expressions easier to work with and understand. Once you express a problem in exponential form, you may often need to simplify further.In the original expression \(c^{2/4}\), the fraction \(\frac{2}{4}\) can be simplified by dividing the numerator and the denominator by their greatest common divisor, which is \(2\). Doing so reduces \(\frac{2}{4}\) to \(\frac{1}{2}\), giving us \(c^{1/2}\).Simplification doesn't change the value of the expression but refines its form to make it more comprehensible. It's similar to simplifying fractions to the lowest terms. This process is not only useful for clarity but essential for solving more complex equations involving exponential terms. The final simplified expression \(c^{1/2}\) directly tells you the result represents the square root of \(c\), making further manipulations straightforward.