What Is The Value Of $\sqrt[3]{512}$?A. 8 B. 6 C. 7 D. 9

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Understanding the Problem

To find the value of $\sqrt[3]{512}$, we need to understand the concept of cube roots and how to calculate them. The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In mathematical terms, if $x = \sqrt[3]{y}$, then $x^3 = y$. We will use this concept to find the value of $\sqrt[3]{512}$.

Breaking Down the Problem

The number 512 can be broken down into its prime factors to make it easier to find its cube root. We can start by finding the prime factors of 512.

Prime Factors of 512

To find the prime factors of 512, we can start by dividing it by the smallest prime number, which is 2.

512 ÷ 2 = 256
256 ÷ 2 = 128
128 ÷ 2 = 64
64 ÷ 2 = 32
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
4 ÷ 2 = 2

Now that we have the prime factors of 512, we can write it as a product of its prime factors.

512 = 2^9

Finding the Cube Root

Now that we have the prime factors of 512, we can find its cube root by taking the cube root of each prime factor.

\sqrt[3]{512} = \sqrt[3]{2^9}

Using the property of exponents that $\sqrt[n]{a^n} = a$, we can simplify the expression.

\sqrt[3]{2^9} = 2^3

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it to find the value of $\sqrt[3]{512}$.

2^3 = 8

Conclusion

Therefore, the value of $\sqrt[3]{512}$ is 8.

Comparison with Options

Let's compare our answer with the options given in the problem.

• A. 8
• B. 6
• C. 7
• D. 9

Our answer, 8, matches option A.

Final Answer

The final answer is $\boxed{8}$.

Frequently Asked Questions

Q: What is the cube root of 512?

A: The cube root of 512 is 8.

Q: How do you find the cube root of a number?

A: To find the cube root of a number, you can break it down into its prime factors and then take the cube root of each prime factor.

Q: What is the property of exponents that we used to simplify the expression?

A: The property of exponents that we used is $\sqrt[n]{a^n} = a$.

Q: How do you evaluate the expression $2^3$?

A: To evaluate the expression $2^3$, you can multiply 2 by itself three times.

Step-by-Step Solution

1. Break down the number 512 into its prime factors.
2. Write the prime factors as a product of their prime factors.
3. Take the cube root of each prime factor.
4. Simplify the expression using the property of exponents $\sqrt[n]{a^n} = a$.
5. Evaluate the expression to find the value of $\sqrt[3]{512}$.

Common Mistakes

• Not breaking down the number into its prime factors.
• Not using the property of exponents to simplify the expression.
• Not evaluating the expression to find the value of $\sqrt[3]{512}$.

Real-World Applications

• Finding the cube root of a number is an important concept in mathematics and has many real-world applications, such as calculating the volume of a cube or the surface area of a cube.
• The concept of cube roots is also used in physics and engineering to calculate the volume and surface area of objects.

Conclusion

In conclusion, finding the value of $\sqrt[3]{512}$ requires breaking down the number into its prime factors, taking the cube root of each prime factor, and then simplifying the expression using the property of exponents. The final answer is $\boxed{8}$.

Q: What is a cube root?

A: A cube root is a value that, when multiplied by itself twice, gives the original number. In mathematical terms, if $x = \sqrt[3]{y}$, then $x^3 = y$.

Q: How do you find the cube root of a number?

A: To find the cube root of a number, you can break it down into its prime factors and then take the cube root of each prime factor. Alternatively, you can use a calculator or a mathematical software to find the cube root.

Q: What is the property of exponents that we used to simplify the expression?

A: The property of exponents that we used is $\sqrt[n]{a^n} = a$. This property allows us to simplify expressions involving cube roots by taking the cube root of each prime factor.

Q: How do you evaluate the expression $2^3$?

A: To evaluate the expression $2^3$, you can multiply 2 by itself three times: $2 \times 2 \times 2 = 8$.

Q: What is the difference between a cube root and a square root?

A: A square root is a value that, when multiplied by itself, gives the original number. A cube root is a value that, when multiplied by itself twice, gives the original number.

Q: Can you give an example of a cube root in real life?

A: Yes, a cube root can be used to calculate the volume of a cube. For example, if you have a cube with a side length of 4, the volume of the cube is $4^3 = 64$. The cube root of 64 is 4, which is the side length of the cube.

Q: How do you find the cube root of a negative number?

A: To find the cube root of a negative number, you can use the property of exponents that $\sqrt[n]{(-a)^n} = -a$. This property allows you to simplify expressions involving cube roots of negative numbers.

Q: Can you give an example of a cube root in physics?

A: Yes, a cube root can be used to calculate the volume of a cube in physics. For example, if you have a cube with a side length of 5, the volume of the cube is $5^3 = 125$. The cube root of 125 is 5, which is the side length of the cube.

Q: How do you find the cube root of a decimal number?

A: To find the cube root of a decimal number, you can use a calculator or a mathematical software. Alternatively, you can use the property of exponents that $\sqrt[n]{a^n} = a$ to simplify the expression.

Q: Can you give an example of a cube root in engineering?

A: Yes, a cube root can be used to calculate the volume of a cube in engineering. For example, if you have a cube with a side length of 6, the volume of the cube is $6^3 = 216$. The cube root of 216 is 6, which is the side length of the cube.

Q: How do you find the cube root of a fraction?

A: To find the cube root of a fraction, you can use the property of exponents that $\sqrt[n]{a^n} = a$ to simplify the expression. Alternatively, you can use a calculator or a mathematical software to find the cube root.

Q: Can you give an example of a cube root in finance?

A: Yes, a cube root can be used to calculate the volume of a cube in finance. For example, if you have a cube with a side length of 7, the volume of the cube is $7^3 = 343$. The cube root of 343 is 7, which is the side length of the cube.

Q: How do you find the cube root of a complex number?

A: To find the cube root of a complex number, you can use the property of exponents that $\sqrt[n]{a^n} = a$ to simplify the expression. Alternatively, you can use a calculator or a mathematical software to find the cube root.

Q: Can you give an example of a cube root in computer science?

A: Yes, a cube root can be used to calculate the volume of a cube in computer science. For example, if you have a cube with a side length of 8, the volume of the cube is $8^3 = 512$. The cube root of 512 is 8, which is the side length of the cube.

Q: How do you find the cube root of a number in a different base?

A: To find the cube root of a number in a different base, you can use the property of exponents that $\sqrt[n]{a^n} = a$ to simplify the expression. Alternatively, you can use a calculator or a mathematical software to find the cube root.

Q: Can you give an example of a cube root in a different base?

A: Yes, a cube root can be used to calculate the volume of a cube in a different base. For example, if you have a cube with a side length of 9 in base 10, the volume of the cube is $9^3 = 729$ in base 10. The cube root of 729 is 9, which is the side length of the cube.

Q: How do you find the cube root of a number in a different base using a calculator?

A: To find the cube root of a number in a different base using a calculator, you can use the calculator's built-in cube root function and specify the base of the number.

Q: Can you give an example of finding the cube root of a number in a different base using a calculator?

A: Yes, to find the cube root of 729 in base 10 using a calculator, you can use the calculator's built-in cube root function and specify the base of the number. The calculator will give you the cube root of 729 in base 10, which is 9.

Q: How do you find the cube root of a number in a different base using a mathematical software?

A: To find the cube root of a number in a different base using a mathematical software, you can use the software's built-in cube root function and specify the base of the number.

Q: Can you give an example of finding the cube root of a number in a different base using a mathematical software?

A: Yes, to find the cube root of 729 in base 10 using a mathematical software, you can use the software's built-in cube root function and specify the base of the number. The software will give you the cube root of 729 in base 10, which is 9.

Q: What is the difference between a cube root and a square root in a different base?

A: A square root in a different base is a value that, when multiplied by itself, gives the original number in that base. A cube root in a different base is a value that, when multiplied by itself twice, gives the original number in that base.

Q: Can you give an example of a cube root in a different base in real life?

A: Yes, a cube root in a different base can be used to calculate the volume of a cube in a different base. For example, if you have a cube with a side length of 10 in base 10, the volume of the cube is $10^3 = 1000$ in base 10. The cube root of 1000 is 10, which is the side length of the cube.

Q: How do you find the cube root of a number in a different base in real life?

A: To find the cube root of a number in a different base in real life, you can use the property of exponents that $\sqrt[n]{a^n} = a$ to simplify the expression. Alternatively, you can use a calculator or a mathematical software to find the cube root.

Q: Can you give an example of a cube root in a different base in physics?

A: Yes, a cube root in a different base can be used to calculate the volume of a cube in a different base in physics. For example, if you have a cube with a side length of 11 in base 10, the volume of the cube is $11^3 = 1331$ in base 10. The cube root of 1331 is 11, which is the side length of the cube.

Q: How do you find the cube root of a number in a different base in physics?

A: To find the cube root of a number in a different base in physics, you can use the property of exponents that $\sqrt[n]{a^n} = a$ to simplify the expression. Alternatively, you can use a calculator or a mathematical software to find the cube root.

Q: Can you give an example of a cube root in a different base in engineering?

A: Yes, a cube root in a different base can be used to calculate the volume of a cube in a different base in engineering. For example, if you have a cube with a side length of 12 in base 10, the volume of the cube is $12^3 = 1728$ in base 10. The cube root of 1728 is 12, which is the side length of the cube.

Q: How do you find the cube root of a number in a different base in engineering?

A: To find the cube root of a number in a different base in engineering, you can use the property of exponents that $\sqrt[n]{a^n} = a$ to simplify the expression. Alternatively,