Applying the Rule of Radicals

Oct 13, 2025 by pinkandtrash.com 128 views

When $$1,250^{\frac{3}{4}}$$ is Written in Its Simplest Radical Form, Which Value Remains Under the Radical?

Understanding the Problem

When dealing with exponents and radicals, it's essential to understand the properties and rules that govern them. In this problem, we're given the expression $$1,250^{\frac{3}{4}}$$ and asked to write it in its simplest radical form. This means we need to simplify the expression by expressing it as a product of a number and a radical.

The Rule of Radicals

The rule of radicals states that for any real number $$a$$ and any positive integer $$n$$, $$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$. This rule allows us to rewrite an expression with a fractional exponent as a radical expression.

Applying the Rule of Radicals

Using the rule of radicals, we can rewrite the expression $$1,250^{\frac{3}{4}}$$ as $$\sqrt[4]{1,250^3}$$. This is because the exponent $$\frac{3}{4}$$ can be rewritten as $$\frac{3}{4} = \frac{3 \times 4}{4 \times 4} = \frac{12}{16}$$, and the denominator $$4$$ is the index of the radical.

Simplifying the Expression

Now that we have rewritten the expression as $$\sqrt[4]{1,250^3}$$, we can simplify it further by evaluating the expression inside the radical. To do this, we need to find the prime factorization of $$1,250$$.

Prime Factorization of 1,250

The prime factorization of $$1,250$$ is $$1,250 = 2 \times 5^4$$. This means that $$1,250$$ can be expressed as a product of its prime factors.

Rewriting the Expression

Using the prime factorization of $$1,250$$, we can rewrite the expression $$\sqrt[4]{1,250^3}$$ as $$\sqrt[4]{(2 \times 5⁴⁾3}$$. This simplifies to $$\sqrt[4]{2^3 \times 5^{12}}$$.

Simplifying the Radical

Now that we have rewritten the expression as $$\sqrt[4]{2^3 \times 5^{12}}$$, we can simplify the radical by evaluating the expression inside the radical. To do this, we need to find the fourth root of $$2^3 \times 5^{12}$$.

Evaluating the Fourth Root

The fourth root of $$2^3 \times 5^{12}$$ is $$\sqrt[4]{2^3} \times \sqrt[4]{5^{12}}$$. This simplifies to $$2^{\frac{3}{4}} \times 5^3$$.

Simplifying the Expression Further

Now that we have rewritten the expression as $$2^{\frac{3}{4}} \times 5^3$$, we can simplify it further by evaluating the expression $$2^{\frac{3}{4}}$$. This is equal to $$\sqrt[4]{2^3}$$, which is equal to $$\sqrt[4]{8}$$.

Evaluating the Fourth Root of 8

The fourth root of $$8$$ is $$\sqrt[4]{8} = 2$$. This means that the value of $$2^{\frac{3}{4}}$$ is $$2$$.

Simplifying the Final Expression

Now that we have evaluated the expression $$2^{\frac{3}{4}}$$, we can simplify the final expression $$2^{\frac{3}{4}} \times 5^3$$. This simplifies to $$2 \times 125$$.

Evaluating the Final Expression

The final expression $$2 \times 125$$ is equal to $$250$$. This means that the value of $$1,250^{\frac{3}{4}}$$ is $$250$$.

Conclusion

In this problem, we were given the expression $$1,250^{\frac{3}{4}}$$ and asked to write it in its simplest radical form. We used the rule of radicals to rewrite the expression as $$\sqrt[4]{1,250^3}$$, and then simplified it further by evaluating the expression inside the radical. We found that the value of $$1,250^{\frac{3}{4}}$$ is $$250$$.

Answer

The value that remains under the radical is 250. However, the question asks for the value that remains under the radical, which is the value that is multiplied by the radical. In this case, the value that remains under the radical is 250 divided by 5 which is 50.
When $$1,250^{\frac{3}{4}}$$ is Written in Its Simplest Radical Form, Which Value Remains Under the Radical? - Q&A

Q: What is the rule of radicals?

A: The rule of radicals states that for any real number $$a$$ and any positive integer $$n$$, $$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$. This rule allows us to rewrite an expression with a fractional exponent as a radical expression.

Q: How do I apply the rule of radicals to the expression $$1,250^{\frac{3}{4}}$$?

A: To apply the rule of radicals, we can rewrite the expression $$1,250^{\frac{3}{4}}$$ as $$\sqrt[4]{1,250^3}$$. This is because the exponent $$\frac{3}{4}$$ can be rewritten as $$\frac{3}{4} = \frac{3 \times 4}{4 \times 4} = \frac{12}{16}$$, and the denominator $$4$$ is the index of the radical.

Q: What is the prime factorization of 1,250?

A: The prime factorization of $$1,250$$ is $$1,250 = 2 \times 5^4$$. This means that $$1,250$$ can be expressed as a product of its prime factors.

Q: How do I rewrite the expression $$\sqrt[4]{1,250^3}$$ using the prime factorization of 1,250?

A: Using the prime factorization of $$1,250$$, we can rewrite the expression $$\sqrt[4]{1,250^3}$$ as $$\sqrt[4]{(2 \times 5⁴⁾3}$$. This simplifies to $$\sqrt[4]{2^3 \times 5^{12}}$$.

Q: How do I simplify the radical $$\sqrt[4]{2^3 \times 5^{12}}$$?

A: To simplify the radical, we need to find the fourth root of $$2^3 \times 5^{12}$$. This can be done by evaluating the expression $$\sqrt[4]{2^3} \times \sqrt[4]{5^{12}}$$.

Q: What is the value of $$\sqrt[4]{2^3}$$?

A: The value of $$\sqrt[4]{2^3}$$ is $$2^{\frac{3}{4}}$$, which is equal to $$\sqrt[4]{8}$$.

Q: What is the value of $$\sqrt[4]{8}$$?

A: The value of $$\sqrt[4]{8}$$ is $$2$$.

Q: How do I simplify the final expression $$2^{\frac{3}{4}} \times 5^3$$?

A: To simplify the final expression, we can evaluate the expression $$2^{\frac{3}{4}}$$, which is equal to $$2$$. Then, we can multiply $$2$$ by $$5^3$$, which is equal to $$125$$.

Q: What is the value of $$2 \times 125$$?

A: The value of $$2 \times 125$$ is $$250$$.

Q: What is the value that remains under the radical?

A: The value that remains under the radical is $$250$$ divided by $$5$$ which is $$50$$.

Conclusion

In this Q&A article, we have discussed the problem of writing the expression $$1,250^{\frac{3}{4}}$$ in its simplest radical form. We have used the rule of radicals to rewrite the expression, and then simplified it further by evaluating the expression inside the radical. We have found that the value of $$1,250^{\frac{3}{4}}$$ is $$250$$, and the value that remains under the radical is $$50$$.

Answer

The value that remains under the radical is 50.