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

Understanding the Problem

When dealing with exponents and roots, it's essential to understand the properties and rules that govern their behavior. In this article, we'll focus on simplifying expressions involving exponents and roots, specifically the expression $\sqrt[5]{1,215}^x$. Our goal is to determine which of the given options is equivalent to this expression.

The Power of Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $a^x$ is equivalent to $a \times a \times a \times \ldots \times a$ (x times). When we have an exponent raised to another exponent, we can simplify it using the rule $(a^x)^y = a^{xy}$. This rule allows us to combine exponents and simplify complex expressions.

The Root of the Problem

Roots, on the other hand, are the inverse operation of exponents. For example, $\sqrt[n]{a}$ is equivalent to $a^{\frac{1}{n}}$. When we have a root raised to another exponent, we can simplify it using the rule $(\sqrt[n]{a})^x = a^{\frac{x}{n}}$. This rule allows us to combine roots and simplify complex expressions.

Simplifying the Given Expression

Now, let's apply these rules to the given expression $\sqrt[5]{1,215}^x$. Using the rule for roots raised to an exponent, we can rewrite this expression as $(1,215)^{\frac{x}{5}}$. This simplification is based on the fact that $\sqrt[n]{a} = a^{\frac{1}{n}}$, so $\sqrt[5]{1,215} = (1,215)^{\frac{1}{5}}$.

Evaluating the Options

Now that we have simplified the given expression, let's evaluate the options:

A. $243^x$ B. $1,215^{\frac{1}{5} x}$ C. $1,215^{\frac{1}{5 x}}$ D. $243^{\frac{1}{x}}$

Option A: $243^x$

Option A is not equivalent to the given expression. The base of the exponent is $243$, whereas the base of the given expression is $1,215$. Therefore, option A is not a correct equivalent expression.

Option B: $1,215^{\frac{1}{5} x}$

Option B is equivalent to the given expression. Using the rule for exponents, we can rewrite this expression as $(1,215)^{\frac{x}{5}}$, which is the same as the simplified expression we obtained earlier.

Option C: $1,215^{\frac{1}{5 x}}$

Option C is not equivalent to the given expression. The exponent is $\frac{1}{5x}$, whereas the exponent of the given expression is $\frac{x}{5}$. Therefore, option C is not a correct equivalent expression.

Option D: $243^{\frac{1}{x}}$

Option D is not equivalent to the given expression. The base of the exponent is $243$, whereas the base of the given expression is $1,215$. Therefore, option D is not a correct equivalent expression.

Conclusion

In conclusion, the correct equivalent expression to $\sqrt[5]{1,215}^x$ is option B: $1,215^{\frac{1}{5} x}$. This expression is equivalent to the given expression because it has the same base and exponent. By applying the rules for exponents and roots, we can simplify complex expressions and determine their equivalent forms.

Final Thoughts

Simplifying exponents and roots is an essential skill in mathematics, and it requires a deep understanding of the underlying rules and properties. By mastering these concepts, you can simplify complex expressions and solve problems with ease. Remember to always apply the rules for exponents and roots, and to simplify expressions by combining like terms and using the properties of exponents and roots.

Common Mistakes to Avoid

When simplifying exponents and roots, it's essential to avoid common mistakes. Here are a few to watch out for:

• Incorrect application of rules: Make sure to apply the rules for exponents and roots correctly. For example, when simplifying a root raised to an exponent, make sure to use the rule $(\sqrt[n]{a})^x = a^{\frac{x}{n}}$.
• Failure to combine like terms: When simplifying expressions, make sure to combine like terms. For example, if you have an expression like $a^x + a^x$, you can combine the terms to get $2a^x$.
• Incorrect simplification of expressions: Make sure to simplify expressions correctly. For example, if you have an expression like $\sqrt[n]{a} \times \sqrt[n]{b}$, you can simplify it to $(ab)^{\frac{1}{n}}$.

Practice Problems

To practice simplifying exponents and roots, try the following problems:

• Simplify the expression $\sqrt[3]{8}^x$.
• Simplify the expression $(\sqrt[4]{16})^y$.
• Simplify the expression $\sqrt[2]{9} \times \sqrt[2]{4}$.

Conclusion

In conclusion, simplifying exponents and roots is an essential skill in mathematics. By mastering these concepts, you can simplify complex expressions and solve problems with ease. Remember to always apply the rules for exponents and roots, and to simplify expressions by combining like terms and using the properties of exponents and roots. With practice and patience, you can become proficient in simplifying exponents and roots and tackle even the most complex problems with confidence.

Understanding the Problem

In our previous article, we explored the concept of simplifying exponents and roots, specifically the expression $\sqrt[5]{1,215}^x$. We determined that the correct equivalent expression is option B: $1,215^{\frac{1}{5} x}$. In this article, we'll answer some frequently asked questions about simplifying exponents and roots.

Q&A

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

A: An exponent is a shorthand way of representing repeated multiplication, while a root is the inverse operation of an exponent. For example, $a^x$ is equivalent to $a \times a \times a \times \ldots \times a$ (x times), while $\sqrt[n]{a}$ is equivalent to $a^{\frac{1}{n}}$.

Q: How do I simplify an expression with a root raised to an exponent?

A: To simplify an expression with a root raised to an exponent, use the rule $(\sqrt[n]{a})^x = a^{\frac{x}{n}}$. For example, $\sqrt[5]{1,215}^x$ can be simplified to $(1,215)^{\frac{x}{5}}$.

Q: What is the rule for combining exponents?

A: The rule for combining exponents is $(a^x)^y = a^{xy}$. This rule allows you to simplify complex expressions by combining like terms.

Q: How do I simplify an expression with multiple roots?

A: To simplify an expression with multiple roots, use the rule $(\sqrt[n]{a}) \times (\sqrt[m]{b}) = (ab)^{\frac{1}{n} \times \frac{1}{m}}$. For example, $\sqrt[2]{9} \times \sqrt[2]{4}$ can be simplified to $(9 \times 4)^{\frac{1}{2} \times \frac{1}{2}} = 36^{\frac{1}{4}}$.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent represents repeated multiplication, while a negative exponent represents repeated division. For example, $a^x$ represents $a \times a \times a \times \ldots \times a$ (x times), while $a^{-x}$ represents $\frac{1}{a} \times \frac{1}{a} \times \frac{1}{a} \times \ldots \times \frac{1}{a}$ (x times).

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, use the rule $a^{-x} = \frac{1}{a^x}$. For example, $a^{-x}$ can be simplified to $\frac{1}{a^x}$.

Common Mistakes to Avoid

When simplifying exponents and roots, it's essential to avoid common mistakes. Here are a few to watch out for:

• Incorrect application of rules: Make sure to apply the rules for exponents and roots correctly. For example, when simplifying a root raised to an exponent, make sure to use the rule $(\sqrt[n]{a})^x = a^{\frac{x}{n}}$.
• Failure to combine like terms: When simplifying expressions, make sure to combine like terms. For example, if you have an expression like $a^x + a^x$, you can combine the terms to get $2a^x$.
• Incorrect simplification of expressions: Make sure to simplify expressions correctly. For example, if you have an expression like $\sqrt[n]{a} \times \sqrt[n]{b}$, you can simplify it to $(ab)^{\frac{1}{n}}$.

Practice Problems

To practice simplifying exponents and roots, try the following problems:

• Simplify the expression $\sqrt[3]{8}^x$.
• Simplify the expression $(\sqrt[4]{16})^y$.
• Simplify the expression $\sqrt[2]{9} \times \sqrt[2]{4}$.

Conclusion

In conclusion, simplifying exponents and roots is an essential skill in mathematics. By mastering these concepts, you can simplify complex expressions and solve problems with ease. Remember to always apply the rules for exponents and roots, and to simplify expressions by combining like terms and using the properties of exponents and roots. With practice and patience, you can become proficient in simplifying exponents and roots and tackle even the most complex problems with confidence.

Final Thoughts

Simplifying exponents and roots is a powerful tool in mathematics, and it requires a deep understanding of the underlying rules and properties. By mastering these concepts, you can simplify complex expressions and solve problems with ease. Remember to always apply the rules for exponents and roots, and to simplify expressions by combining like terms and using the properties of exponents and roots. With practice and patience, you can become proficient in simplifying exponents and roots and tackle even the most complex problems with confidence.