Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[4]{256} $$

Step-by-Step Solution

Answer

\( \sqrt[4]{256} = 4 \)

Step 1: Identify the given expression

The expression we need to simplify is \( \sqrt[4]{256} \). This represents the fourth root of 256.

Step 2: Recognize the form of 256

256 is a perfect power of 2 because it can be expressed as \( 2^8 \). This will help in taking the fourth root.

Step 3: Apply the fourth root

To find \( \sqrt[4]{256} \), we determine the number when raised to the power of 4 gives us 256. Here this number is 2 since \( 2^4 = 16 \) and \( 16^2 = 256 \), so \( \sqrt[4]{256} = 2^2 = 4 \).

Step 4: Conclude the solution

Therefore, the simplified form of \( \sqrt[4]{256} \) is 4.

Key Concepts

Fourth root

Imagine you have a number and you want to find which other number, when multiplied by itself four times, will give you that original number. This process is finding the "fourth root". So, for the number 256, we need to find a number, let's call it \( x \), such that \( x \times x \times x \times x = 256 \).
This is particularly handy when simplifying radicals, as recognizing the fourth root can convert a complex radical into a simpler number. In the case of \( \sqrt[4]{256} \), the fourth root of 256 is 4 since \( 4^4 = 256 \).
Remember, fourth roots can apply to both positive and negative numbers (though with more complexity), but typically we focus on the positive result in basic exercises for simplicity.

Perfect power

When we talk about a "perfect power", we're referring to a number that can be written as another number raised to an integer power. It's akin to having a perfect square, but works for any integer power, like perfect cubes, or in our case, perfect fourth powers.
For instance, 256 is a perfect power because it can be expressed as \( 2^8 \). This discovery helps when working with larger radicals as we can break down the number into powers.

• This kind of simplification is useful because it can reduce a seemingly complicated radical into a simpler one.
• For example, simplifying \( \sqrt[4]{2^8} \) is easier if you recognize it as \( 2^2 \).

Identifying perfect powers allows for swift simplification, enabling students to tackle complex expressions with greater ease.

Radical expression

A radical expression involves roots of any degree. The fourth root, square root, or cube root are all types of radicals. These expressions are usually marked with the symbol \( \sqrt{} \), but the degree of the root is also indicated when it's not a square root.
When you come across \( \sqrt[4]{256} \), it's telling you to find the fourth root. Working with radicals can sometimes appear daunting because the expressions under the root sign may be large.

• But breaking down the numbers into manageable parts, like finding perfect powers, makes this easier.
• This skill is crucial when you need the result in a simpler form, such as in other study problems or exams.

Understanding radical expressions helps simplify math equations, which can be essential for algebra or calculus scenarios.

Rationalization

In algebra, rationalizing usually means getting rid of radicals from the denominator of a fraction. It simplifies calculations and it's often needed to arrive at the simplest form of an answer.
We didn't have a denominator in our original expression, but if we did and it involved a radical, we would multiply the numerator and denominator by a term that would eliminate the radical from the denominator.

• This is often done because it avoids complications with dividing by irrational numbers.
• It makes the expression easier to work with in subsequent calculations or when combining fractions.

Rationalization is a standard method used to simplify expressions, ensuring everything is neat and orderly for further arithmetic operations.