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Chapter 8: Problem 31

Simplify. $$\sqrt[3]{192}+5 \sqrt[3]{3}$$

Short Answer

Expert verified
Simplified to \ 9 \sqrt[3]{3}.

Step by step solution

01

Identify the terms

The expression consists of two cube root terms: \ \ \ \( \sqrt[3]{192} \) and \( 5 \sqrt[3]{3} \).
02

Factorize inside the cube root

Factorize \(192\) inside the cube root to find a suitable simplification: \[192 = 64 \cdot 3 \] which gives \[\sqrt[3]{192} = \sqrt[3]{64 \cdot 3} = \sqrt[3]{64} \sqrt[3]{3} = 4 \sqrt[3]{3} \] because \(\sqrt[3]{64} = 4\).
03

Combine like terms

Now that we have \(4\sqrt[3]{3}\) from \(\sqrt[3]{192}\), combine it with the other term: \ \ \ \(4 \sqrt[3]{3} + 5 \sqrt[3]{3}\) = \ \ 9 \sqrt[3]{3}.

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Key Concepts

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

cube roots
A cube root of a number is a value that, when multiplied by itself three times (cubed), gives the original number. For example, the cube root of 8 is 2 because 2 脳 2 脳 2 = 8. In mathematical notation, the cube root is represented by the radical symbol with an index of 3, such as \( \sqrt[3]{8} \). To simplify or manipulate cube roots, it's helpful to recognize perfect cubes like 1, 8, 27, 64, and so on. These perfect cubes make factorization, a subsequent concept, much easier when simplifying expressions.
factorization
Factorization involves breaking down a number into its prime factors or factors that are easier to work with in a given problem. For instance, in our exercise, we factorize 192 inside the cube root as \( 192 = 64 \cdot 3 \). That's because 64 is a perfect cube \( (4 \times 4 \times 4) \). Identifying 64 lets us take its cube root easily, helping to simplify the radical expression. Without recognizing such factorization, simplifying cube roots becomes much more difficult.
combining like terms
To simplify expressions, look for 'like terms'鈥攖erms that have the same radical part. In this case, after factorizing, we find our terms are \(4 \sqrt[3]{3}\) and \(5 \sqrt[3]{3}\). Both terms have a common radical part \( \sqrt[3]{3}\), allowing us to combine them. Combining like terms involves adding or subtracting their coefficients. Here, \(4 \sqrt[3]{3} \) combined with \(5 \sqrt[3]{3}\) results in \(9 \sqrt[3]{3}\). This final expression is a simplified form of the original problem.
radical expressions
Radical expressions involve roots, such as square roots or cube roots. In the given exercise, we deal with cube roots, noted as \( \sqrt[3]{...} \). Simplifying these expressions often starts with factorization to identify perfect cubes, followed by combining any like terms. Working with radicals can initially seem challenging, but recognizing patterns and properties鈥攕uch as those of perfect squares or cubes鈥攈elps streamline the simplification process. Mastery of these concepts ensures solving these expressions becomes a straightforward task.

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