Which Is Equivalent To $\sqrt[5]{1,215}^x$?A. $243^x$B. $1,215^{\frac{1}{5} X}$C. $1,215^{\frac{1}{5x}}$D. $243^{\frac{1}{x}}$

Introduction

In mathematics, simplifying exponential expressions is a crucial skill that helps us to manipulate and solve complex equations. One of the key concepts in this area is the ability to rewrite expressions in equivalent forms. In this article, we will explore how to simplify the expression $\sqrt[5]{1,215}^x$ and find its equivalent form.

Understanding the Expression

The given expression is $\sqrt[5]{1,215}^x$. To simplify this expression, we need to understand the properties of exponents and radicals. The expression can be rewritten as $(1,215)^{\frac{1}{5}x}$ using the property of radicals that $\sqrt[n]{a} = a^{\frac{1}{n}}$. This is a fundamental concept in mathematics that helps us to simplify expressions involving radicals.

Simplifying the Expression

Now that we have rewritten the expression as $(1,215)^{\frac{1}{5}x}$, we can simplify it further. To do this, we need to find the prime factorization of 1,215. The prime factorization of 1,215 is $3^5 \cdot 5 \cdot 7$. Using this factorization, we can rewrite the expression as $(3^5 \cdot 5 \cdot 7)^{\frac{1}{5}x}$.

Using the Property of Exponents

Now that we have rewritten the expression as $(3^5 \cdot 5 \cdot 7)^{\frac{1}{5}x}$, we can use the property of exponents that $(a^m)^n = a^{mn}$ to simplify it further. Applying this property, we get $3^{5 \cdot \frac{1}{5}x} \cdot 5^{\frac{1}{5}x} \cdot 7^{\frac{1}{5}x}$.

Simplifying the Expression Further

Now that we have simplified the expression to $3^{x} \cdot 5^{\frac{1}{5}x} \cdot 7^{\frac{1}{5}x}$, we can see that it is equivalent to $3^x \cdot (5 \cdot 7)^{\frac{1}{5}x}$. Using the property of exponents that $a^m \cdot a^n = a^{m+n}$, we can simplify this expression further to $3^x \cdot 35^{\frac{1}{5}x}$.

Finding the Equivalent Form

Now that we have simplified the expression to $3^x \cdot 35^{\frac{1}{5}x}$, we can see that it is equivalent to $243^{\frac{1}{x}}$. To see why this is the case, we need to understand the properties of exponents and radicals. The expression $243^{\frac{1}{x}}$ can be rewritten as $(3^5)^{\frac{1}{x}}$ using the property of exponents that $(a^m)^n = a^{mn}$. This is equivalent to $3^{5 \cdot \frac{1}{x}}$, which is the same as $3^{\frac{5}{x}}$.

Conclusion

In conclusion, the expression $\sqrt[5]{1,215}^x$ is equivalent to $243^{\frac{1}{x}}$. This is because we can rewrite the expression as $(1,215)^{\frac{1}{5}x}$ using the property of radicals that $\sqrt[n]{a} = a^{\frac{1}{n}}$. We can then simplify this expression further using the property of exponents that $(a^m)^n = a^{mn}$. Finally, we can see that the expression is equivalent to $243^{\frac{1}{x}}$ by understanding the properties of exponents and radicals.

Answer

Q: What is the property of radicals that helps us to simplify expressions involving radicals?

A: The property of radicals that helps us to simplify expressions involving radicals is $\sqrt[n]{a} = a^{\frac{1}{n}}$. This property allows us to rewrite expressions involving radicals in terms of exponents.

Q: How do we simplify the expression $\sqrt[5]{1,215}^x$?

A: To simplify the expression $\sqrt[5]{1,215}^x$, we need to understand the properties of exponents and radicals. We can rewrite the expression as $(1,215)^{\frac{1}{5}x}$ using the property of radicals that $\sqrt[n]{a} = a^{\frac{1}{n}}$. We can then simplify this expression further using the property of exponents that $(a^m)^n = a^{mn}$.

Q: What is the prime factorization of 1,215?

A: The prime factorization of 1,215 is $3^5 \cdot 5 \cdot 7$.

Q: How do we use the property of exponents to simplify the expression $(3^5 \cdot 5 \cdot 7)^{\frac{1}{5}x}$?

A: We can use the property of exponents that $(a^m)^n = a^{mn}$ to simplify the expression $(3^5 \cdot 5 \cdot 7)^{\frac{1}{5}x}$. This gives us $3^{5 \cdot \frac{1}{5}x} \cdot 5^{\frac{1}{5}x} \cdot 7^{\frac{1}{5}x}$.

Q: How do we simplify the expression $3^{x} \cdot 5^{\frac{1}{5}x} \cdot 7^{\frac{1}{5}x}$ further?

A: We can simplify the expression $3^{x} \cdot 5^{\frac{1}{5}x} \cdot 7^{\frac{1}{5}x}$ further by using the property of exponents that $a^m \cdot a^n = a^{m+n}$. This gives us $3^x \cdot 35^{\frac{1}{5}x}$.

Q: How do we find the equivalent form of the expression $3^x \cdot 35^{\frac{1}{5}x}$?

A: We can find the equivalent form of the expression $3^x \cdot 35^{\frac{1}{5}x}$ by understanding the properties of exponents and radicals. The expression $3^x \cdot 35^{\frac{1}{5}x}$ is equivalent to $243^{\frac{1}{x}}$.

Q: What is the correct answer to the original question?

A: The correct answer to the original question is B. $1,215^{\frac{1}{5} x}$.

Q: Why is the expression $\sqrt[5]{1,215}^x$ equivalent to $243^{\frac{1}{x}}$?

A: The expression $\sqrt[5]{1,215}^x$ is equivalent to $243^{\frac{1}{x}}$ because we can rewrite the expression as $(1,215)^{\frac{1}{5}x}$ using the property of radicals that $\sqrt[n]{a} = a^{\frac{1}{n}}$. We can then simplify this expression further using the property of exponents that $(a^m)^n = a^{mn}$. Finally, we can see that the expression is equivalent to $243^{\frac{1}{x}}$ by understanding the properties of exponents and radicals.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill that helps us to manipulate and solve complex equations. By understanding the properties of exponents and radicals, we can rewrite expressions involving radicals in terms of exponents and simplify them further. The correct answer to the original question is B. $1,215^{\frac{1}{5} x}$.