Problem 10 - A Graphical Approach to College Algebra

Question

Write each expression in radical form. Assume that all variables represent positive real numbers. $$p^{3 / 4}$$

Step-by-Step Solution

Answer

\(p^{3/4} = \sqrt[4]{p^3}\)

Step 1: Identify the Exponent and Radicand

The expression given is \(p^{3/4}\). This is in the form of \(p^{m/n}\) where \(m\) is the numerator and \(n\) is the denominator. For this expression, the exponent \(m = 3\) and the root \(n = 4\).

Step 2: Write in Radical Form

Using the rule that \(a^{m/n} = \sqrt[n]{a^m}\), rewrite \(p^{3/4}\) as a radical. Here, \(p^{3/4} = \sqrt[4]{p^3}\). This means you take the fourth root of \(p^3\).

Step 3: Final Radical Form

Confirm that \(p^{3/4}\) is correctly rewritten as \(\sqrt[4]{p^3}\) using the rules of exponents and radicals: the denominator \(4\) becomes the index of the radical, and the numerator \(3\) becomes the power of the radicand inside the radical.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It's about repeated multiplication. For example, if the base is 2 and the exponent is 3, written as \(2^3\), it means multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).
Exponents can be integers, like in the example above, but they can also be fractions, where additional rules apply. Fractional exponents show both powers and roots together.
When dealing with variables, like in the expression \(p^{3/4}\), the exponential rules remain the same, representing growth or enlargement. Always identify the base (\(p\) here) and the exponent (\(3/4\)) to understand what the operation involves.

Fourth Root

The fourth root of a number or a variable, like in \(\sqrt[4]{x}\), means finding a number that, when multiplied by itself four times, equals \(x\). Fourth roots extend the idea of square roots (multiplied by themselves twice) and cube roots (multiplied by themselves thrice).
The process of finding a fourth root can be thought of as the reverse of exponentiation, specifically raising to the fourth power.

• For example, the fourth root of 16 is 2 because \(2 \times 2 \times 2 \times 2 = 16\).
• In contrast, for variables, the concept remains the same: \(\sqrt[4]{p^3}\) seeks a value which multiplied by itself four times gives \(p^3\).

Understanding roots helps in simplifying expressions and finding solutions to equations involving powers.

Rational Exponents

Rational exponents, like \(p^{3/4}\), combine powers and roots through fractional numbers. They are an alternative form to radical expressions. The numerator of the fraction (\(3\) in this case) indicates the power, while the denominator (\(4\)) represents the root.
Here's how to interpret it:

• The expression \(p^{3/4}\) tells you to first find the fourth root of \(p\), and then raise the result to the third power.
• This can equally be viewed by raising \(p\) to the third power and then taking the fourth root.

Both approaches will give you the same outcome, thanks to the properties of rational exponents.
Using rational exponents can simplify computations and make expressions easier to work with, especially when dealing with complex numbers or variables.