Simplify: 125 4 3 125^{\frac{4}{3}} 12 5 3 4 ​ Options: A. 625 B. 3,125 C. 20 D. 100

Understanding Exponents and Radicals

In mathematics, exponents and radicals are two fundamental concepts that are used to represent repeated multiplication and roots of numbers. Exponents are used to represent repeated multiplication of a number, while radicals are used to represent roots of numbers. In this article, we will focus on simplifying the expression $125^{\frac{4}{3}}$ using exponent rules and radical properties.

What is an Exponent?

An exponent is a small number that is written above and to the right of a number or a variable. It represents the number of times the base number or variable is multiplied by itself. For example, in the expression $2^3$, the exponent 3 represents the number of times the base number 2 is multiplied by itself. In this case, $2^3 = 2 \times 2 \times 2 = 8$.

What is a Radical?

A radical is a symbol that is used to represent the root of a number. The most common radical is the square root, which is represented by the symbol $\sqrt{ }$. For example, $\sqrt{16} = 4$ because 4 is the number that, when multiplied by itself, gives 16.

Simplifying Exponents with Radicals

Now that we have a basic understanding of exponents and radicals, let's focus on simplifying the expression $125^{\frac{4}{3}}$. To simplify this expression, we need to use the rule of exponents that states $(a^m)^n = a^{m \times n}$. In this case, we can rewrite the expression as $(125^{\frac{1}{3}})^4$.

Finding the Cube Root of 125

To simplify the expression $(125^{\frac{1}{3}})^4$, we need to find the cube root of 125. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In this case, the cube root of 125 is 5, because $5 \times 5 \times 5 = 125$.

Simplifying the Expression

Now that we have found the cube root of 125, we can simplify the expression $(125^{\frac{1}{3}})^4$. Using the rule of exponents, we can rewrite the expression as $5^4$. To evaluate this expression, we need to multiply 5 by itself four times: $5 \times 5 \times 5 \times 5 = 625$.

Conclusion

In this article, we have simplified the expression $125^{\frac{4}{3}}$ using exponent rules and radical properties. We have found the cube root of 125, which is 5, and then used the rule of exponents to simplify the expression. The final answer is $5^4 = 625$.

Answer

The correct answer is A. 625.

Additional Examples

Here are some additional examples of simplifying exponents with radicals:

• $64^{\frac{3}{2}} = (64^{\frac{1}{2}})^3 = 8^3 = 512$
• $27^{\frac{2}{3}} = (27^{\frac{1}{3}})^2 = 3^2 = 9$
• $16^{\frac{4}{2}} = (16^{\frac{1}{2}})^4 = 4^4 = 256$

Frequently Asked Questions

In this article, we will answer some frequently asked questions related to simplifying the expression $125^{\frac{4}{3}}$.

Q: What is the cube root of 125?

A: The cube root of 125 is 5, because $5 \times 5 \times 5 = 125$.

Q: How do I simplify the expression $125^{\frac{4}{3}}$?

A: To simplify the expression $125^{\frac{4}{3}}$, you need to use the rule of exponents that states $(a^m)^n = a^{m \times n}$. In this case, you can rewrite the expression as $(125^{\frac{1}{3}})^4$.

Q: What is the value of $5^4$?

A: The value of $5^4$ is 625, because $5 \times 5 \times 5 \times 5 = 625$.

Q: Can I simplify the expression $125^{\frac{4}{3}}$ using a different method?

A: Yes, you can simplify the expression $125^{\frac{4}{3}}$ using a different method. One way to do this is to rewrite the expression as $(125^{\frac{1}{3}})^4 = (5)^4 = 625$.

Q: What is the relationship between exponents and radicals?

A: Exponents and radicals are two fundamental concepts in mathematics that are used to represent repeated multiplication and roots of numbers. Exponents are used to represent repeated multiplication of a number, while radicals are used to represent roots of numbers.

Q: How do I evaluate the expression $5^4$?

A: To evaluate the expression $5^4$, you need to multiply 5 by itself four times: $5 \times 5 \times 5 \times 5 = 625$.

Q: Can I use a calculator to simplify the expression $125^{\frac{4}{3}}$?

A: Yes, you can use a calculator to simplify the expression $125^{\frac{4}{3}}$. However, it's always a good idea to understand the underlying math and use a calculator as a check.

Q: What is the final answer to the expression $125^{\frac{4}{3}}$?

A: The final answer to the expression $125^{\frac{4}{3}}$ is 625.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying the expression $125^{\frac{4}{3}}$. We have discussed the cube root of 125, the rule of exponents, and how to evaluate the expression $5^4$. We have also provided additional examples and tips for simplifying exponents with radicals.

Additional Tips

Here are some additional tips for simplifying exponents with radicals:

• Always start by simplifying the expression inside the parentheses.
• Use the rule of exponents to rewrite the expression.
• Evaluate the expression by multiplying the base number by itself the required number of times.
• Check your answer using a calculator.

By following these tips and understanding the underlying math, you can simplify exponents with radicals with ease.