Which Of The Following Choices Are Equivalent To The Expression X 8 / 15 X^{8 / 15} X 8/15 ? Check All That Apply.A. ( X 5 ) 1 / 8 \left(x^5\right)^{1 / 8} ( X 5 ) 1/8 B. X 8 5 \sqrt[5]{x^8} 5 X 8 ​ C. X 5 8 \sqrt[8]{x^5} 8 X 5 ​ D. ( X 5 ) 8 (\sqrt[5]{x})^8 ( 5 X ​ ) 8 E.

When dealing with exponents, it's essential to understand the rules and properties that govern them. One of the most critical concepts is the ability to rewrite expressions in equivalent forms. In this article, we'll explore the choices provided and determine which ones are equivalent to the expression $x^{8 / 15}$.

The Power of Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $x^3$ is equivalent to $x \cdot x \cdot x$. When dealing with fractions, we can use the rule $\left(x^m\right)^n = x^{m \cdot n}$ to simplify expressions.

Choice A: $\left(x^5\right)^{1 / 8}$

Let's start by analyzing Choice A. We can rewrite this expression using the rule $\left(x^m\right)^n = x^{m \cdot n}$. In this case, $m = 5$ and $n = 1/8$. Therefore, we can simplify the expression as follows:

$$\left(x^5\right)^{1 / 8} = x^{5 \cdot 1/8} = x^{5/8}$$

This expression is not equivalent to $x^{8 / 15}$, so we can eliminate Choice A.

Choice B: $\sqrt[5]{x^8}$

Next, let's examine Choice B. We can rewrite this expression using the rule $\sqrt[n]{x^m} = x^{m/n}$. In this case, $n = 5$ and $m = 8$. Therefore, we can simplify the expression as follows:

$$\sqrt[5]{x^8} = x^{8/5}$$

This expression is not equivalent to $x^{8 / 15}$, so we can eliminate Choice B.

Choice C: $\sqrt[8]{x^5}$

Now, let's analyze Choice C. We can rewrite this expression using the rule $\sqrt[n]{x^m} = x^{m/n}$. In this case, $n = 8$ and $m = 5$. Therefore, we can simplify the expression as follows:

$$\sqrt[8]{x^5} = x^{5/8}$$

This expression is not equivalent to $x^{8 / 15}$, so we can eliminate Choice C.

Choice D: $(\sqrt[5]{x})^8$

Next, let's examine Choice D. We can rewrite this expression using the rule $\left(x^m\right)^n = x^{m \cdot n}$. In this case, $m = 1/5$ and $n = 8$. Therefore, we can simplify the expression as follows:

$$(\sqrt[5]{x})^8 = \left(x^{1/5}\right)^8 = x^{8/5}$$

This expression is not equivalent to $x^{8 / 15}$, so we can eliminate Choice D.

Conclusion

After analyzing each choice, we can conclude that none of the expressions provided are equivalent to $x^{8 / 15}$. However, we can use the rule $\left(x^m\right)^n = x^{m \cdot n}$ to rewrite the original expression as follows:

$$x^{8/15} = \left(x^{1/3}\right)^{8/5}$$

This expression is equivalent to the original expression, but it's not among the choices provided.

Final Answer

In this article, we'll address some common questions related to exponents and equivalent expressions.

Q: What is the rule for rewriting exponents?

A: The rule for rewriting exponents is $\left(x^m\right)^n = x^{m \cdot n}$. This rule allows us to simplify expressions by multiplying the exponents.

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

A: To simplify an expression with a fraction exponent, you can use the rule $\left(x^m\right)^n = x^{m \cdot n}$. For example, if you have the expression $x^{3/4}$, you can rewrite it as $\left(x^3\right)^{1/4}$.

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

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the reciprocal of the base is raised to a power. For example, $x^3$ is equivalent to $x \cdot x \cdot x$, while $x^{-3}$ is equivalent to $1/x^3$.

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

A: To simplify an expression with a negative exponent, you can use the rule $x^{-m} = 1/x^m$. For example, if you have the expression $x^{-2}$, you can rewrite it as $1/x^2$.

Q: What is the rule for rewriting radicals?

A: The rule for rewriting radicals is $\sqrt[n]{x^m} = x^{m/n}$. This rule allows us to simplify expressions by dividing the exponent by the index of the radical.

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

A: To simplify an expression with a radical, you can use the rule $\sqrt[n]{x^m} = x^{m/n}$. For example, if you have the expression $\sqrt[3]{x^4}$, you can rewrite it as $x^{4/3}$.

Q: What is the relationship between exponents and radicals?

A: Exponents and radicals are related in that they both represent repeated multiplication. Exponents are used to represent repeated multiplication of a single base, while radicals are used to represent repeated multiplication of a single base with a fractional exponent.

Q: How do I convert an expression with a radical to an expression with an exponent?

A: To convert an expression with a radical to an expression with an exponent, you can use the rule $\sqrt[n]{x^m} = x^{m/n}$. For example, if you have the expression $\sqrt[3]{x^4}$, you can rewrite it as $x^{4/3}$.

Q: How do I convert an expression with an exponent to an expression with a radical?

A: To convert an expression with an exponent to an expression with a radical, you can use the rule $x^{m/n} = \sqrt[n]{x^m}$. For example, if you have the expression $x^{4/3}$, you can rewrite it as $\sqrt[3]{x^4}$.

Conclusion

In this article, we've addressed some common questions related to exponents and equivalent expressions. We've covered topics such as rewriting exponents, simplifying expressions with fraction exponents, and converting between exponents and radicals. By understanding these concepts, you'll be better equipped to simplify complex expressions and solve problems involving exponents and radicals.