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Chapter 9: Problem 494

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

Short Answer

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

Step by step solution

01

- 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 \(\root3{81} - \root3{192} = 3 \root3{3} - 4 \root3{3} = -\root3{3} \).
02

- 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 \(\root4{512} - \root4{32} = 4 \root4{2} - 2 \root4{2} = 2 \root4{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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鈥檚 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.

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