Problem 494 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\quad \sqrt[3]{81}-\sqrt[3]{192}\) \(\sqrt[4]{512}-\sqrt[4]{32}\)

Step-by-Step Solution

Answer

\(-\root3{3} \text{ and } 2 \root4{2} \)

Step 1 - Simplify cube roots

Simplify \(\root3{81} \text{ and } \root3{192} \). The prime factorizations are 81 = 3^4 and 192 = 2^6 \. \(\root3{81} = \root3{3^4} = 3^{4/3} \) and \(\root3{192} = \root3{2^6} = 2^{6/3} = 2^2 = 4\). Convert them: \(\root3{81} = \root3{27 \times 3} = 3 \root3{3} \) and \(\root3{192} = \root3{64 \times 3} = 4 \root3{3} \). So \(\root3{81} - \root3{192} = 3 \root3{3} - 4 \root3{3} = -\root3{3} \).

Step 2 - Simplify fourth roots

Simplify \(\root4{512} \text{ and } \root4{32} \). The prime factorizations are 512 = 2^9 and 32 = 2^5 \. \(\root4{512} = \root4{2^9} = 2^{9/4} \) and \(\root4{32} = \root4{2^5} = 2^{5/4} \). Convert them: \(\root4{512} = \root4{256 \times 2} = 4 \root4{2} \) and \(\root4{32} = \root4{16 \times 2} = 2 \root4{2} \). So \(\root4{512} - \root4{32} = 4 \root4{2} - 2 \root4{2} = 2 \root4{2} \).

Key Concepts

Cube Roots

Cube roots deal with finding a number that, when multiplied by itself three times, gives the original number. For instance, the cube root of 27 is 3 because \[3 \times 3 \times 3 = 27\]. Understanding cube roots can make solving more complex problems easier.
Certain steps help simplify cube roots:

• Find the prime factorization of the number. For example, the prime factors of 81 are \[3^4\].
• Write the number under the cube root in terms of its prime factors. So, \[\root3{81} = \root3{3^4}\] can be rewritten as \[3 \times \root3{3}\].
• If there are multiples of three in the exponents, simplify them. Like in our example, \[\root3{81} = 3 \root3{3}\].

These steps show the transformation of complicated numbers into simpler, easy-to-handle forms.

Fourth Roots

Fourth roots involve finding a number that multiplies by itself four times to form the original number. For instance, the fourth root of 16 is 2 since \[2 \times 2 \times 2 \times 2 = 16\]. Simplifying fourth roots requires a different approach compared to cube roots.
Here’s how to simplify fourth roots:

• Perform the prime factorization of the number. For example, 512 can be broken into \[2^9\].
• Express the number under the fourth root in terms of its prime factors. So, \[\root4{512}\] becomes \[\root4{2^9}\].
• Look for multiples of four in the exponents to simplify. In our example, \[\root4{512} = 2^{9/4}\], which simplifies further if there are rational parts, like convert the fraction part.

These steps make dealing with fourth roots systematic and less daunting.

Prime Factorization

Prime factorization breaks down a number into its basic prime factors. Prime numbers only have two distinct positive divisors: one and themselves.
It’s crucial for simplifying radical expressions:

• To find the prime factors of a number, start dividing by the smallest prime (2, if the number is even).
• Continue dividing by prime numbers (like 3, 5, 7, etc.) until you reach a prime.
• For example, the prime factorization of 192 is \[2^6 \times 3\].

Prime factorization assists in understanding and simplifying cube and fourth roots by breaking numbers into more manageable parts.

Radical Simplification

Radical simplification involves reducing expressions under a radical sign to their simplest form. This makes them easier to work with and understand.
Steps to simplify radicals include:

• Use prime factorization to break down the number inside the radical.
• Check for pairs (or triples, or quads in the case of cube and fourth roots) of factors that can come out of the radical.
• Simplify what's left inside the radical and multiply by what was extracted. For instance, \[\root3{81} = 3 \root3{3}\].

These steps help in transforming complex radical expressions into simpler forms, making further calculations or integrations more manageable.