Cube root of a negative number (example) | Pre-Algebra | Khan Academy

TL;DR

The cube root of negative 512 is -8.

Transcript

We are asked to find the cube root of negative 512. Or another way to think about it is if I have some number, and it is equal to the cube root of negative 512, this just means that if I take that number and I raise it to the third power, then I get negative 512. And if it doesn't jump out at you immediately what this is the cube of, or what we hav... Read More

Key Insights

• 🫚 The cube root of a negative number involves breaking it down into the cube root of -1 multiplied by the cube root of the positive number.
• ✊ The cube root of -1 is -1 because -1 raised to the power of 3 equals -1.
• 🫚 The prime factorization of a number helps determine its cube root.
• 🧑‍🏭 Grouping the factors can simplify the calculation and reveal the cube root.

Questions & Answers

Q: How can we find the cube root of negative 512?

To find the cube root of negative 512, we can break it down into the cube root of -1 multiplied by the cube root of 512. The cube root of -1 is -1, and the prime factorization of 512 is 2 to the power of 9.

Q: How do we determine the cube root of -1?

The cube root of -1 is -1 because when you raise -1 to the third power, you get -1. This can be verified by multiplying -1 by itself three times.

Q: What is the prime factorization of 512?

The prime factorization of 512 is 2 to the power of 9, which means it can be expressed as 2 multiplied by itself nine times. The factors of 2 are obtained by dividing 512 repeatedly by 2 until reaching 1.

Q: Why do we group the twos when finding the cube root of 512?

We group the twos in order to simplify the cube root of 512. By grouping them into three sets of twos, we can express 512 as 8 raised to the power of 3. This simplifies the calculation.

Summary & Key Takeaways

• The cube root of negative 512 can be expressed as the cube root of -1 multiplied by the cube root of 512.

• The cube root of -1 is -1, and the prime factorization of 512 shows that it can be written as 2 to the power of 9.

• By grouping the twos into three sets, we find that the cube root of 512 is 8.

• Therefore, the cube root of negative 512 is -8.