Problem 72 - Intermediate Algebra

Question

Simplify by combining like radicals. All variables represent positive real numbers. $$ \sqrt[4]{48}-\sqrt[4]{243}-\sqrt[4]{768} $$

Step-by-Step Solution

Answer

The simplified expression is \( -5 \cdot \sqrt[4]{3} \).

Step 1: Prime Factorization

First, find the prime factorization of each number under the fourth root to simplify: - 48 = 2^4 * 3 - 243 = 3^5 - 768 = 2^8 * 3 We will use these factorizations in the next step.

Step 2: Express Each Root Using Prime Factors

Express each expression using the fourth root of the prime factorization:- \( \sqrt[4]{48} = \sqrt[4]{2^4 \cdot 3} = 2 \cdot \sqrt[4]{3} \)- \( \sqrt[4]{243} = \sqrt[4]{3^5} = 3 \cdot \sqrt[4]{3} \)- \( \sqrt[4]{768} = \sqrt[4]{2^8 \cdot 3} = 2^2 \cdot \sqrt[4]{3} = 4 \cdot \sqrt[4]{3} \)

Step 3: Combine Like Radicals

Now, combine the like radicals (all terms have \( \sqrt[4]{3} \)):- \( 2 \cdot \sqrt[4]{3} - 3 \cdot \sqrt[4]{3} - 4 \cdot \sqrt[4]{3} \)This simplifies to:- \( (2 - 3 - 4) \cdot \sqrt[4]{3} = -5 \cdot \sqrt[4]{3} \)

Step 4: Conclusion

The simplified expression of the given radicals is:\( -5 \cdot \sqrt[4]{3} \).

Key Concepts

Prime Factorization

Prime factorization is a technique used to express a number as a product of its prime numbers. Understanding and using prime factors can make simplifying complex expressions easier.
For example, if you take the number 48, you can break it into prime factors like so:

• 48 can be divided by 2 (the smallest prime number), resulting in 24.
• 24 can be further divided by 2 to get 12.
• Divide 12 by 2 to get 6.
• Divide 6 by 2 to obtain 3, which is a prime number.

Thus, 48 can be expressed as a product of prime factors: \( 2^4 \times 3 \). The purpose of prime factorization in problems like our exercise is to simplify the numbers under roots, making the next steps more manageable.

Fourth Roots

The fourth root of a number is a value that, when raised to the power of 4, returns the original number. It's like finding a number's square root, but one step further.
For our exercise, to simplify the expressions with fourth roots, we rely on the prime factorization of the number.

• For instance, to simplify \( \sqrt[4]{48} \), using the factorization \( 2^4 \times 3 \), we can extract the fourth root of \( 2^4 \).
• This results in 2, as \( 2^4 \) under a fourth root equals 2.
• The remaining \( \sqrt[4]{3} \) stays under the root since 3 is prime and not raised to the power of 4.

This same process applies to the other numbers in our expression, allowing us to simplify each root effectively by using prime factorization.

Combining Like Terms

Combining like terms involves simplifying expressions by adding or subtracting terms that have identical components. In the context of radical expressions, terms are alike if they contain the same root factor.
In our exercise, each radical term shares \( \sqrt[4]{3} \) as a common factor:

• \( 2 \cdot \sqrt[4]{3} \)
• \( -3 \cdot \sqrt[4]{3} \)
• \( -4 \cdot \sqrt[4]{3} \)

To combine like terms, you just add or subtract their coefficients while keeping the root factor constant:

• Combine coefficients: \( 2 - 3 - 4 = -5 \)
• The simplified expression is \( -5 \cdot \sqrt[4]{3} \)

By reducing the expression this way, we achieve a simpler form and clear up the complexity of the initial problem.