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Chapter 10: Problem 78

Simplify. Assume that all variables represent positive real numbers. \(-\sqrt[3]{64 y^{18}}\)

Short Answer

Expert verified
-4 y^6

Step by step solution

01

- Factor inside the root

Start by factoring the number inside the root. Notice that 64 can be written as \(64 = 4^3\). Rewrite the expression as \[-\sqrt[3]{4^3 \times y^{18}}\]
02

- Apply the cube root

Since we have the cube root, apply the cube root to each factor separately: \[-\sqrt[3]{4^3} \times \sqrt[3]{y^{18}} = -4 \times y^{18/3} = -4 \times y^6\]
03

- Simplify the expression

Combine the results from the previous step to obtain the simplified expression: \[-4y^6\]

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Key Concepts

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

Cube Roots
Understanding cube roots is essential. A cube root of a number is a value that, when multiplied by itself three times, equals the original number. For instance, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\). In our exercise, we notice that 64 can be expressed as \(4^3\). So, the cube root of 64 simplifies directly to 4. This concept allows us to break down complex expressions under the cube root into something more manageable.
Factoring
Factoring is the key technique to simplify expressions under a root. Factoring means breaking down a number or an expression into its building blocks, or factors, that when multiplied together give the original number or expression. Let's look at the number 64 again. We know \(64 = 4^3\). This simplifies the cube root of 64. Similarly, when dealing with variables like \(y^{18}\), it helps to express the exponent in terms of the cube root. Here, \(18\) is expressed as \(6 \times 3\), so factoring it as \(y^{3 \times 6}\) simplifies the cube root further. By breaking down each part, simplification becomes straightforward.
Variables in Exponents
Variables in exponents can look intimidating, but they're manageable with the right approach. When simplifying expressions with variables in exponents under a root, rewrite the exponent to align with the root. In our given exercise, we have \(y^{18}\) under a cube root. Recognize that \(18 = 3 \times 6\), and you can treat the variable part separately as \(y^{18/3} = y^6\). This means taking the cube root of \(y^{18}\) gives \(y^6\). By treating variables step-by-step, it's easier to simplify expressions correctly.

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