Problem 39 Simplify by factoring. $$ \s... [FREE SOLUTION]

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Chapter 10: Problem 39

Simplify by factoring. $$ \sqrt[3]{-16 x^{6}} $$

Short Answer

Expert verified

-2x^2 \root[3]{2}

Step by step solution

Identify the expression inside the cube root

The given expression is $\root[3]{-16 x^{6}}$. First, observe what can be factored inside the cube root.

Factor the radicand

$-16 x^6$ can be factored as $-16 = -8 \times 2$ and $x^6 = (x^2)^3$, so the expression becomes $\root[3]{-8 \times 2 \times (x^2)^3}$.

Simplify using cube root properties

We can use the property $\root[k]{a \times b} = \root[k]{a} \times \root[k]{b}$. Applying this, we get: $\root[3]{-8} \times \root[3]{2} \times \root[3]{(x^2)^3}$.

Calculate the individual cube roots

The cube root of $-8$ is $-2$, the cube root of $2$ is $\root[3]{2}$, and the cube root of $(x^2)^3$ is $x^2$. Therefore, we have: $-2 \times \root[3]{2} \times x^2$.

Combine the results

Multiply the terms together to get the simplified expression: $-2x^2 \root[3]{2}$.

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Key Concepts

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

Factoring

Factoring is a crucial step in simplifying expressions. It involves breaking down a complex expression into simpler parts that are multiplied together. This can be especially useful when dealing with roots, particularly cube roots. For example, in the expression $\root[3]{-16 x^{6}}$, we first need to factor both the constant term and the variable term. By recognizing that $-16 = -8 \times 2$ and $x^6 = (x^2)^3$, we effectively prepare the expression for further simplification. Always look for factors that are perfect cubes, as these will simplify more easily under a cube root.

Cube Root Properties

Cube roots have specific properties that can help simplify expressions. One important property is that the cube root of a product is equal to the product of the cube roots: $\root[3]{a \times b} = \root[3]{a} \times \root[3]{b}$. This is crucial in the given exercise. After factoring $-16 x^6$ into $-8 \times 2 \times (x^2)^3$, we use this property to break it down as follows: $\root[3]{-8} \times \root[3]{2} \times \root[3]{(x^2)^3}$. Each individual term can then be simplified separately. For instance, $\root[3]{-8} = -2$, and $\root[3]{(x^2)^3} = x^2$. These steps make the expression much easier to manage.

Radicand

The term 'radicand' refers to the number or expression inside the radical sign (in this case, the cube root). In $\root[3]{-16 x^{6}}$, the radicand is \

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91影视

Simplify by factoring. $$ \sqrt[3]{-16 x^{6}} $$

Short Answer

Step by step solution

Identify the expression inside the cube root

Factor the radicand

Simplify using cube root properties

Calculate the individual cube roots

Combine the results

Key Concepts

Factoring

Cube Root Properties

Radicand

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