Problem 93 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt[4]{208 m^{4} n} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 2m \sqrt[4]{13n} \).

Step 1: Identify the Expression

First, we are given the expression \( \sqrt[4]{208 m^{4} n} \). This is a fourth root expression involving multiplication inside the radical.

Step 2: Factor Inside the Radical

Factor the number 208 into its prime factors. We have \( 208 = 2^4 \times 13 \). Thus, the expression becomes \( \sqrt[4]{2^4 \times 13 \times m^4 \times n} \).

Step 3: Simplify Perfect Fourth Powers

Identify perfect fourth powers inside the radical. Both \( 2^4 \) and \( m^4 \) are perfect fourth powers. The fourth root of these are \( 2 \) and \( m \), respectively. Therefore, \( \sqrt[4]{2^4 \times m^4} = 2m \).

Step 4: Combine and Simplify the Expression

Pull the perfect fourth powers out from under the radical, and leave the rest under the radical: \( 2m \sqrt[4]{13n} \). The expression simplifies to \( 2m \sqrt[4]{13n} \) by combining terms.

Key Concepts

Fourth Roots

In mathematics, roots are a fundamental concept that allow us to find a value which, when multiplied by itself a certain number of times, gives the original number. When talking about the fourth root, it means we're looking for a number that, when multiplied by itself four times, equals the original number inside the radical sign. Let's break this down further:

• The fourth root of a number \( x \) is written as \( \sqrt[4]{x} \).
• It involves finding a number \( a \) so that \( a^4 = x \).

Taking the example from our exercise, \( \sqrt[4]{208 m^4 n} \), we aim to simplify what's under the fourth root symbol by finding parts of the expression that are perfect fourth powers. These can then be taken out of the radical, simplifying the expression.

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. Each number can be broken down this way, which simplifies operations under radical expressions.

To simplify \( \sqrt[4]{208 m^4 n} \) from the exercise, we need the prime factors of 208. By dividing the number 208 by the smallest prime numbers, we get:

• 208 divided by 2 gives us 104.
• 104 divided by 2 gives us 52.
• 52 divided by 2 gives us 26.
• 26 divided by 2 gives us 13, and 13 itself is a prime number.

This process results in \( 208 = 2^4 \times 13 \). Substituting back, \( 208 \) becomes \( 2^4 \times 13 \), which helps in identifying perfect fourth powers in the expression.

Exponents

Exponents are a mathematical shorthand for expressing repeated multiplication of a number by itself. In the expression \( \sqrt[4]{208 m^4 n} \), exponents display the number of times a number is multiplied by itself. Here’s what you need to consider:

• Each base number with an exponent represents its base multiplied by itself as many times as the exponent shows, like \( m^4 = m \times m \times m \times m \).
• When simplifying, you look for exponents that match the index of the root. For a fourth root, a perfect fourth power is any expression where the exponent is a multiple of 4, which simplifies nicely under a fourth root.

In this context, \( m^4 \) is a perfect fourth power, and its fourth root is \( m \). That is why \( 2^4 \) and \( m^4 \) simplify directly to 2 and \( m \), respectively, outside of the radical.

Perfect Fourth Powers

Perfect fourth powers are numbers that can be expressed as the fourth power of integers. Recognizing these helps in simplifying fourth roots.

A perfect fourth power takes the form \( a^4 \), where \( a \) remains as it is when taken under the fourth root. Let’s explore this with our exercise:

• The expression \( 2^4 \) fits perfectly under a fourth root because it implies \( (2)^4 \) equals 16, and the fourth root of 16 is simply 2.
• The same reasoning applies to \( m^4 \), making \( m \) the fourth root of \( m^4 \).

When these expressions are simplified, they emerge out of the radical, which simplifies \( \sqrt[4]{208 m^4 n} \) to \( 2m \sqrt[4]{13n} \). Any term that isn't a perfect fourth power, like \( 13n \), remains under the radical, completing the simplification.