Problem 19 - Intermediate Algebra

Question

Simplify each expression. $$ \sqrt[4]{48} $$

Step-by-Step Solution

Answer

\( \sqrt[4]{48} = 2 \sqrt[4]{3} \).

Step 1: Prime Factorization of 48

To simplify \( \sqrt[4]{48} \), begin by doing the prime factorization of 48. 48 can be factored as follows: 48 \( = 2^4 \times 3 \).

Step 2: Apply the 4th Root to Each Factor

Next, take the 4th root of each factor. Since \( 48 = 2^4 \times 3 \), it follows that \( \sqrt[4]{48} = \sqrt[4]{2^4 \times 3} \).

Step 3: Simplify the Powers

Using the property of radicals where \( \sqrt[a]{b^a} = b \), we simplify \( \sqrt[4]{2^4} \) to 2. Therefore, \( \sqrt[4]{48} = \sqrt[4]{2^4} \times \sqrt[4]{3} = 2 \times \sqrt[4]{3} \).

Step 4: Write the Final Simplified Expression

Now, the expression is simplified as much as possible: \( 2 \times \sqrt[4]{3} \). Therefore, the simplified form of \( \sqrt[4]{48} \) is \( 2 \sqrt[4]{3} \).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, the prime factors of 48 are derived by dividing by the smallest prime numbers until we cannot divide any further.

• Start by dividing 48 by 2, the smallest prime number: 48 divided by 2 equals 24.
• Divide 24 by 2 again: 24 divided by 2 equals 12.
• Continue: 12 divided by 2 equals 6.
• One more division: 6 divided by 2 equals 3, a prime number itself.

Hence, the prime factorization of 48 is given by multiplying all the prime numbers together: \[ 48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3. \]Breaking larger numbers into their prime components makes it easier to work with them, especially when simplifying radical expressions.

Properties of Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, and nth roots. Understanding the properties of radicals is crucial when simplifying expressions like \( \sqrt[4]{48} \). Here are some key properties:

• Product Property: The nth root of a product is the product of the nth roots. For example, \( \sqrt[n]{a \times b} = \sqrt[n]{a} \times \sqrt[n]{b} \).
• Power Property: When the root corresponds with the exponent, they can cancel each other out, such as \( \sqrt[a]{b^a} = b \).
• Quotient Property: Similar to the product property, \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \), provided \( b eq 0 \).
• Simplifying Radicals: If an expression under the radical sign can be written as the power of another expression, it can be simplified by reducing the powers, using the above properties to break down complex radicals into more manageable expressions.

In our exercise, using the power property helps simplify \( \sqrt[4]{2^4} \) to 2, turning the expression into \( 2 \sqrt[4]{3} \). Understanding these properties makes working with radicals more efficient and less daunting.

Nth Roots

Nth roots generalize the concept of square and cube roots to any positive integer n. The nth root of a number \( x \) is a number \( y \) such that \( y^n = x \). For example, the 4th root of 16 is 2 because \( 2^4 = 16 \). Here’s how nth roots work and why they matter:

• Notation: The nth root of \( x \) is written as \( \sqrt[n]{x} \), where n is a positive integer called the "index."
• Whole Powers: When the number under the root is a perfect nth power, the nth root is an integer. For instance, \( \sqrt[4]{16} \) is 2, because 16 is \( 2^4 \).
• Non-perfect Powers: If the number isn't a perfect nth power, the result is usually an irrational number, which can often be simplified but not completely eliminated (for instance, \( \sqrt[4]{3} \) can't be simplified further with integers).
• Simplifying: To simplify expressions with nth roots, work on breaking the radicand into factors and reducing them step by step using root properties.

In the case of \( \sqrt[4]{48} \), we first expressed 48 in terms of its prime factors, then extracted any whole nth roots (\( 2^4 \)) to simplify the overall expression to \( 2 \sqrt[4]{3} \). Understanding nth roots enables deeper manipulation and transformation of expressions, ideal for solving complex mathematical problems.