Problem 63 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ 12^{\frac{5}{4}} $$

Step-by-Step Solution

Answer

The radical form of \(12^{\frac{5}{4}}\) is \((\sqrt[4]{12})^5\).

Step 1: Understand the Problem

The problem asks us to write the expression \(12^{\frac{5}{4}}\) as a radical expression in its simplest form. This involves converting the expression from exponential form to radical form.

Step 2: Identify the Exponential Form

The given exponential expression is \(12^{\frac{5}{4}}\), where the base is 12, the numerator of the exponent is 5, and the denominator is 4.

Step 3: Convert the Denominator to a Radical

The denominator of the exponent (4) indicates the root to be taken, which is the fourth root in this case. Therefore, \(12^{\frac{1}{4}} = \sqrt[4]{12}\).

Step 4: Apply the Numerator as a Power

Now, apply the numerator of the exponent (5) as a power to the radical expression. This transforms the expression into \((\sqrt[4]{12})^5\).

Step 5: Simplify the Radical Expression

The expression \(12^{\frac{5}{4}}\) as a radical is \((\sqrt[4]{12})^5\). This is the simplest radical form for the given expression, as we cannot simplify \(\sqrt[4]{12}\) further without decimal approximations, which we avoid in simplest radical form.

Key Concepts

Exponential Form

Exponential form is a way of expressing numbers using a base and an exponent. It allows us to represent repeated multiplication of a number efficiently. When we write something like \(12^{\frac{5}{4}}\), the number 12 is the base, and the fraction \(\frac{5}{4}\) is the exponent. The exponent tells us two things:

• The denominator (4) indicates the degree of the root we need to take.
• The numerator (5) tells us the power to which the base is raised after the root is taken.

This way, the exponential form combines roots and powers into one concise notation. By understanding the structure of the exponent, we can effectively convert between exponential and radical forms.

Simplest Radical Form

Giving an expression in its simplest radical form means expressing it in terms of roots, while simplifying as much as possible without using decimal approximations. For instance, with \(12^{\frac{5}{4}}\), the goal is to express the number in terms of roots and powers without unnecessary complexity. Converting it into a radical expression involves rewriting the expression using a root corresponding to the denominator, and then raising that rooted number to an appropriate power.In many cases, simplifying further would involve finding perfect squares, cubes, or fourth powers within the radicand (the number under the root). However, sometimes, as in this case, a radical expression such as \(\sqrt[4]{12}\) cannot be simplified further without altering its exactness.

Fourth Root

Roots are the opposite of powers and help in finding simpler forms of expressions in mathematics. The fourth root indicates the number which, when multiplied by itself four times, gives the original number.In our problem, the expression \(12^{\frac{1}{4}}\) translates to the fourth root of 12, written as \(\sqrt[4]{12}\) in radical form. This step is essential for converting exponential expressions into radical terms.Mathematically, finding a root can generally be a bit challenging as not every number is a perfect power of 4. In some cases, as with 12, the fourth root results in an irrational number, leading us to keep the root symbol rather than converting to a potentially less accurate decimal approximation. Understanding fourth roots is key when working with fractional exponents.