Problem 62 Simplify $\sqrt[3]{-250}$... [FREE SOLUTION]

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Chapter 10: Problem 62

Simplify $\sqrt[3]{-250}$

Short Answer

Expert verified

-5 \cdot \sqrt[3]{2}

Step by step solution

Understand the Problem

The task is to simplify the expression $\sqrt[3]{-250}$. We are looking for the cube root of -250.

Identify the Cube Root of Negative Numbers

Remember that the cube root of a negative number is also a negative number. So we will find the cube root of 250 and then take the negative of that result.

Express 250 as a Product of its Prime Factors

Find the prime factors of 250. We get 250 = 2 * 5^3.

Simplify the Cube Root

We can write: $\sqrt[3]{250} = \sqrt[3]{2 \cdot 5^3}$. Since $\sqrt[3]{5^3} = 5$, the expression simplifies to $\sqrt[3]{2} \cdot 5$.

Include the Negative Sign

Since we are finding the cube root of -250, we need to include the negative sign. Therefore, $\sqrt[3]{-250} = -5 \cdot \sqrt[3]{2}$.

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Key Concepts

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

cube root

A cube root is the number that, when multiplied by itself three times, gives the original number. For instance, the cube root of 8 is 2 since 2 * 2 * 2 = 8. We denote the cube root of a number as $\sqrt[3]{a}$. Unlike square roots, cube roots can be taken of both positive and negative numbers. This is because a negative number multiplied by itself three times will yield a negative result. For example, \$-2 \cdot -2 \cdot -2 = -8\$$.

negative numbers

When dealing with cube roots of negative numbers, it's important to note that the result will also be negative. This is different from square roots, which cannot have a negative result within the set of real numbers. For example, $\sqrt[3]{-8} = -2$ because \$-2 \cdot -2 \cdot -2 = -8\$$. In our exercise, simplifying $\sqrt[3]{-250}$ means finding the cube root of 250 first and then applying the negative sign.

prime factorization

Prime factorization is the process of breaking a number down into its smallest prime factors. For example, the prime factorization of 250 is done by dividing it by prime numbers:\
\$250 \div 2 = 125\$$ \$125 \div 5 = 25\$$ \$25 \div 5 = 5\$$ \$5 \div 5 = 1\$$
Thus, we have that 250 = $\2 \cdot 5^3$. This allows us to further simplify the cube root.

simplification process

The simplification process involves rewriting a complex expression in its simplest form. To simplify $\sqrt[3]{250} = \sqrt[3]{2 \cdot 5^3}$ we recognize that the cube root of $\5^3$ is simply 5, as $\sqrt[3]{5^3} = 5$. We then include the negative sign, because we are dealing with $\sqrt[3]{-250}$. So, we get $\-5 \cdot \sqrt[3]{2}$. This results in our simplified form: $\-5 \cdot \sqrt[3]{2}$.

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91影视

Simplify \(\sqrt[3]{-250}\)

Short Answer

Step by step solution

Understand the Problem

Identify the Cube Root of Negative Numbers

Express 250 as a Product of its Prime Factors

Simplify the Cube Root

Include the Negative Sign

Key Concepts

cube root

negative numbers

prime factorization

simplification process

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