Rewrite The Expression Without An Exponent Or Radical: 125 1 3 125^{\frac{1}{3}} 12 5 3 1 Show Your Work Here:Hint: To Add The Square Root Symbol ( □ (\sqrt{\square} ( □ ], Type root.

Mar 11, 2025 by ADMIN 190 views

Rewrite the Expression without an Exponent or Radical: $125^{\frac{1}{3}}$

Understanding the Problem

The given expression is $125^{\frac{1}{3}}$ . This expression involves a fractional exponent, which can be rewritten as a radical. To rewrite the expression without an exponent or radical, we need to understand the properties of exponents and radicals.

Properties of Exponents and Radicals

A fractional exponent can be rewritten as a radical using the following property:

$a^{\frac{m}{n}} = \sqrt[n]{a^m}$

where $a$ is the base, $m$ is the exponent, and $n$ is the index of the radical.

Rewriting the Expression

Using the property above, we can rewrite the expression $125^{\frac{1}{3}}$ as:

$\sqrt[3]{125^1}$

Simplifying the Expression

Now, we can simplify the expression by evaluating the exponent:

$\sqrt[3]{125^1} = \sqrt[3]{125}$

Finding the Cube Root

To find the cube root of 125, we need to find a number that, when multiplied by itself three times, equals 125.

Step 1: Find the Cube Root of 125

The cube root of 125 is a number that, when multiplied by itself three times, equals 125. We can find this number by trial and error or by using a calculator.

Step 2: Evaluate the Cube Root

Using a calculator, we can evaluate the cube root of 125:

$\sqrt[3]{125} = 5$

Conclusion

Therefore, the expression $125^{\frac{1}{3}}$ can be rewritten as:

$\boxed{5}$

Understanding the Result

The result of $125^{\frac{1}{3}}$ is 5, which is the cube root of 125. This result can be verified by cubing 5:

$5^3 = 125$

Real-World Applications

The concept of fractional exponents and radicals has many real-world applications, including:

Physics: In physics, fractional exponents are used to describe the motion of objects with variable acceleration.
Engineering: In engineering, fractional exponents are used to describe the behavior of complex systems.
Computer Science: In computer science, fractional exponents are used in algorithms for solving mathematical problems.

Conclusion

In conclusion, the expression $125^{\frac{1}{3}}$ can be rewritten as 5, which is the cube root of 125. This result can be verified by cubing 5. The concept of fractional exponents and radicals has many real-world applications, including physics, engineering, and computer science.
Q&A: Rewrite the Expression without an Exponent or Radical

Frequently Asked Questions

Q: What is the difference between an exponent and a radical?

A: An exponent is a small number that is raised to a power, while a radical is a symbol that represents a root of a number. For example, $2^3$ is an exponent, while $\sqrt{4}$ is a radical.

Q: How do I rewrite an expression with a fractional exponent as a radical?

A: To rewrite an expression with a fractional exponent as a radical, you can use the following property:

$a^{\frac{m}{n}} = \sqrt[n]{a^m}$

where $a$ is the base, $m$ is the exponent, and $n$ is the index of the radical.

Q: How do I rewrite the expression $125^{\frac{1}{3}}$ without an exponent or radical?

A: To rewrite the expression $125^{\frac{1}{3}}$ without an exponent or radical, you can use the property above:

$\sqrt[3]{125^1} = \sqrt[3]{125}$

Q: What is the cube root of 125?

A: The cube root of 125 is a number that, when multiplied by itself three times, equals 125. We can find this number by trial and error or by using a calculator.

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

A: To find the cube root of a number, you can use a calculator or try different numbers until you find one that, when multiplied by itself three times, equals the original number.

Q: What are some real-world applications of fractional exponents and radicals?

A: Fractional exponents and radicals have many real-world applications, including:

Physics: In physics, fractional exponents are used to describe the motion of objects with variable acceleration.
Engineering: In engineering, fractional exponents are used to describe the behavior of complex systems.
Computer Science: In computer science, fractional exponents are used in algorithms for solving mathematical problems.

Q: How do I simplify an expression with a radical?

A: To simplify an expression with a radical, you can try to find a perfect square or cube that can be factored out of the radical.

Q: What is a perfect square or cube?

A: A perfect square is a number that can be expressed as the product of an integer and itself, such as 4 or 9. A perfect cube is a number that can be expressed as the product of an integer and itself three times, such as 8 or 27.

Q: How do I factor out a perfect square or cube from a radical?

A: To factor out a perfect square or cube from a radical, you can use the following property:

$\sqrt{a^2} = a$

$\sqrt[3]{a^3} = a$

where $a$ is the number inside the radical.

Q: What are some common mistakes to avoid when working with radicals?

A: Some common mistakes to avoid when working with radicals include:

Not simplifying the radical: Make sure to simplify the radical by factoring out any perfect squares or cubes.
Not using the correct property: Make sure to use the correct property for rewriting an expression with a fractional exponent as a radical.
Not checking the answer: Make sure to check the answer by plugging it back into the original expression.

Conclusion

In conclusion, rewriting an expression without an exponent or radical requires understanding the properties of exponents and radicals. By using the correct property and simplifying the radical, you can rewrite an expression without an exponent or radical. Remember to check the answer by plugging it back into the original expression.