Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without radicals or exponents.. $$ \sqrt[4]{256} $$

Step-by-Step Solution

Answer

The simplified number without radicals or exponents is 4.

Step 1: Rewrite the fourth root as a power

We can rewrite the given number as a power with the base 256 and exponent 1/4, like this: $$ \sqrt[4]{256} = 256^{\frac{1}{4}} $$

Step 2: Find the equivalent rational exponent

To find the equivalent rational exponent, we need to determine the value of 256 raised to the power of 1/4. In other words, we are looking for a number (let's call it n) such that: $$ n^4 = 256 $$ The fourth root of 256 is 4 because \(4^4 = 256\).

Step 3: Simplify the number without radicals or exponents

We can now replace the variable n in the equation with the found value of 4: $$ 256^{\frac{1}{4}} = 4 $$ So the simplified number without radicals or exponents is 4.

Key Concepts

Radicals

In mathematics, radicals refer to expressions that involve roots, such as square roots or cube roots. A radical is typically denoted by the radical sign (√) followed by a number or expression beneath or alongside it.
Radicals are another way to represent roots of numbers, and they can often be transformed into expressions with rational exponents for simpler calculations.
For example:

• The square root of a number is written as \( \sqrt{n} \), which is equivalent to raising the number to the power of \( \frac{1}{2} \).
• Similarly, the cube root of a number can be written as \( \sqrt[3]{n} \) or \( n^{\frac{1}{3}} \).

Understanding radicals is essential because they provide a powerful tool for solving equations, simplifying expressions, and performing arithmetic operations on roots.
It's noteworthy that while radicals and rational exponents are different notations, they can be interchangeably used for easier manipulation of mathematical expressions.

Fourth Root

The fourth root of a number is a special type of radical that involves finding a number which, when raised to the power of four, equals the original number.
It is represented by the notation \( \sqrt[4]{n} \). For instance, to find the fourth root of 256, you need to determine what number, when multiplied by itself four times, equals 256.
Let's see how this works step-by-step:

• The expression \( \sqrt[4]{256} \) is seeking a solution to the equation \( n^4 = 256 \).
• By testing integers, you find that \( 4^4 = 256 \), so 4 is the fourth root of 256.

Sometimes, converting the fourth root to an exponent form such as \( 256^{\frac{1}{4}} \) can simplify calculations, especially when using calculators or algebraic manipulations.
This concept demonstrates that radicals can be expressed as rational exponents, providing an alternative way of solving and simplifying problems.

Exponentiation

Exponentiation refers to the mathematical operation involving numbers known as the base and the exponent. This operation is often used to simplify multiplication by expressing repeated multiplication as a single expression.
The base is the number being multiplied, and the exponent denotes how many times the base is used as a factor.
For example:

• In the expression \( 256^{\frac{1}{4}} \), 256 is the base, and \( \frac{1}{4} \) is the exponent.
• This means you are finding what number, when raised to the fourth power, equals 256.
• Through exponentiation, you discover that \( 4^4 = 256 \).

Understanding exponentiation is crucial in various areas of mathematics as it is widely used in calculus, physics, and computer science for calculations involving growth, decay, and scalability.
In the context of rational exponents, it provides a bridge between radicals and exponential expressions, ensuring students can seamlessly switch perspectives when solving complex mathematical tasks.