Simplify The Following Expression: − 32 3 \sqrt[3]{-32} 3 − 32 ​

Introduction

In mathematics, simplifying expressions is an essential skill that helps us solve problems efficiently and accurately. One of the most common types of expressions that require simplification is radicals, particularly cube roots. In this article, we will focus on simplifying the expression $\sqrt[3]{-32}$.

Understanding Cube Roots

Before we dive into simplifying the expression, let's first understand what cube roots are. A cube root of a number is a value that, when multiplied by itself twice, gives the original number. In other words, if $x$ is the cube root of $y$, then $x^3 = y$. The cube root symbol is denoted by $\sqrt[3]{x}$.

Simplifying the Expression

To simplify the expression $\sqrt[3]{-32}$, we need to find the cube root of $-32$. We can start by factoring $-32$ into its prime factors. We know that $-32 = -2^5$. Since the cube root of $-2^5$ is $-2^{\frac{5}{3}}$, we can rewrite the expression as $\sqrt[3]{-32} = -2^{\frac{5}{3}}$.

Simplifying the Exponent

Now that we have the expression in the form $-2^{\frac{5}{3}}$, we can simplify the exponent. To do this, we need to find the cube root of $2^5$. We know that the cube root of $2^5$ is $2^{\frac{5}{3}}$. Therefore, we can rewrite the expression as $\sqrt[3]{-32} = -2^{\frac{5}{3}} = -2 \cdot 2^{\frac{2}{3}}$.

Simplifying the Expression Further

We can simplify the expression further by evaluating the cube root of $2^{\frac{2}{3}}$. We know that the cube root of $2^{\frac{2}{3}}$ is $2^{\frac{2}{3} \cdot \frac{1}{3}} = 2^{\frac{2}{9}}$. Therefore, we can rewrite the expression as $\sqrt[3]{-32} = -2 \cdot 2^{\frac{2}{3}} = -2 \cdot 2^{\frac{2}{9}}$.

Final Simplification

We can simplify the expression further by evaluating the product of $-2$ and $2^{\frac{2}{9}}$. We know that the product of $-2$ and $2^{\frac{2}{9}}$ is $-2 \cdot 2^{\frac{2}{9}} = -2^{\frac{11}{9}}$. Therefore, we can rewrite the expression as $\sqrt[3]{-32} = -2^{\frac{11}{9}}$.

Conclusion

In this article, we simplified the expression $\sqrt[3]{-32}$ by factoring the number $-32$ into its prime factors, simplifying the exponent, and evaluating the cube root of the resulting expression. We found that the simplified expression is $-2^{\frac{11}{9}}$. This example demonstrates the importance of simplifying expressions in mathematics and how it can help us solve problems efficiently and accurately.

Frequently Asked Questions

• What is the cube root of $-32$?
• How do you simplify the expression $\sqrt[3]{-32}$?
• What is the final simplified expression for $\sqrt[3]{-32}$?

Step-by-Step Solution

1. Factor the number $-32$ into its prime factors.
2. Simplify the exponent by finding the cube root of $2^5$.
3. Evaluate the cube root of $2^{\frac{2}{3}}$.
4. Simplify the expression further by evaluating the product of $-2$ and $2^{\frac{2}{9}}$.
5. Rewrite the expression in the final simplified form.

Additional Resources

• Cube Root Formula
• Simplifying Expressions
• Prime Factorization

Related Articles

• Simplify the Expression: $\sqrt[3]{-27}$
• Simplify the Expression: $\sqrt[3]{-64}$
• Simplify the Expression: $\sqrt[3]{-125}$
  

Introduction

In our previous article, we simplified the expression $\sqrt[3]{-32}$ by factoring the number $-32$ into its prime factors, simplifying the exponent, and evaluating the cube root of the resulting expression. In this article, we will answer some frequently asked questions related to simplifying cube roots.

Q&A

Q: What is the cube root of $-27$?

A: The cube root of $-27$ is $-3$ because $(-3)^3 = -27$.

Q: How do you simplify the expression $\sqrt[3]{-64}$?

A: To simplify the expression $\sqrt[3]{-64}$, we need to find the cube root of $-64$. We know that $-64 = -2^6$, so we can rewrite the expression as $\sqrt[3]{-64} = -2^2 = -4$.

Q: What is the final simplified expression for $\sqrt[3]{-125}$?

A: To simplify the expression $\sqrt[3]{-125}$, we need to find the cube root of $-125$. We know that $-125 = -5^3$, so we can rewrite the expression as $\sqrt[3]{-125} = -5$.

Q: How do you simplify the expression $\sqrt[3]{-216}$?

A: To simplify the expression $\sqrt[3]{-216}$, we need to find the cube root of $-216$. We know that $-216 = -6^3$, so we can rewrite the expression as $\sqrt[3]{-216} = -6$.

Q: What is the cube root of $-343$?

A: The cube root of $-343$ is $-7$ because $(-7)^3 = -343$.

Q: How do you simplify the expression $\sqrt[3]{-729}$?

A: To simplify the expression $\sqrt[3]{-729}$, we need to find the cube root of $-729$. We know that $-729 = -9^3$, so we can rewrite the expression as $\sqrt[3]{-729} = -9$.

Q: What is the final simplified expression for $\sqrt[3]{-1000}$?

A: To simplify the expression $\sqrt[3]{-1000}$, we need to find the cube root of $-1000$. We know that $-1000 = -10^3$, so we can rewrite the expression as $\sqrt[3]{-1000} = -10$.

Conclusion

In this article, we answered some frequently asked questions related to simplifying cube roots. We provided step-by-step solutions to each question and explained the reasoning behind each answer. We hope that this article has been helpful in understanding how to simplify cube roots.

Frequently Asked Questions

• What is the cube root of $-27$?
• How do you simplify the expression $\sqrt[3]{-64}$?
• What is the final simplified expression for $\sqrt[3]{-125}$?
• How do you simplify the expression $\sqrt[3]{-216}$?
• What is the cube root of $-343$?
• How do you simplify the expression $\sqrt[3]{-729}$?
• What is the final simplified expression for $\sqrt[3]{-1000}$?

Step-by-Step Solution

1. Factor the number into its prime factors.
2. Simplify the exponent by finding the cube root of the resulting expression.
3. Evaluate the cube root of the resulting expression.
4. Rewrite the expression in the final simplified form.

Additional Resources

• Cube Root Formula
• Simplifying Expressions
• Prime Factorization

Related Articles

• Simplify the Expression: $\sqrt[3]{-32}$
• Simplify the Expression: $\sqrt[3]{-27}$
• Simplify the Expression: $\sqrt[3]{-64}$