Which Expression Is Equivalent To $\left(x^{\frac{1}{4}} Y^{16}\right)^{\frac{1}{2}}$?A. $x^{\frac{1}{2}} Y^4$ B. $x^{\frac{1}{8}} Y^8$ C. $x^{\frac{1}{4}} Y^8$ D. $x^{\frac{1}{4}} Y^4$

$$

Understanding the Problem

The given expression is a combination of two variables, x and y, raised to certain powers. We are asked to simplify this expression and find an equivalent form from the given options. To solve this problem, we need to apply the rules of exponents and understand the properties of radicals.

Applying the Rules of Exponents

When we have an expression in the form of $\left(a^m b^n\right)^p$, we can simplify it by applying the rule of exponents, which states that $\left(a^m b^n\right)^p = a^{mp} b^{np}$. In this case, we have $\left(x^{\frac{1}{4}} y^{16}\right)^{\frac{1}{2}}$, and we can apply the rule of exponents to simplify it.

Simplifying the Expression

Using the rule of exponents, we can simplify the expression as follows:

$\left(x^{\frac{1}{4}} y^{16}\right)^{\frac{1}{2}} = x^{\frac{1}{4} \times \frac{1}{2}} y^{16 \times \frac{1}{2}}$

Evaluating the Exponents

Now, we can evaluate the exponents to simplify the expression further:

$x^{\frac{1}{4} \times \frac{1}{2}} = x^{\frac{1}{8}}$

$y^{16 \times \frac{1}{2}} = y^8$

Combining the Terms

Now that we have simplified the expression, we can combine the terms to get the final result:

$x^{\frac{1}{8}} y^8$

Comparing with the Options

Now, we can compare our result with the given options to find the equivalent expression:

A. $x^{\frac{1}{2}} y^4$ B. $x^{\frac{1}{8}} y^8$ C. $x^{\frac{1}{4}} y^8$ D. $x^{\frac{1}{4}} y^4$

Conclusion

Based on our simplification, we can see that the equivalent expression is:

B. $x^{\frac{1}{8}} y^8$

This is the correct answer, and we can conclude that the expression $\left(x^{\frac{1}{4}} y^{16}\right)^{\frac{1}{2}}$ is equivalent to $x^{\frac{1}{8}} y^8$.

Final Answer

The final answer is B. $x^{\frac{1}{8}} y^8$.

Understanding Exponents and Radicals

Exponents and radicals are fundamental concepts in mathematics that help us simplify complex expressions. In the previous article, we discussed how to simplify the expression $\left(x^{\frac{1}{4}} y^{16}\right)^{\frac{1}{2}}$. In this article, we will answer some frequently asked questions about simplifying expressions with exponents and radicals.

Q: What is the rule of exponents?

A: The rule of exponents states that $\left(a^m b^n\right)^p = a^{mp} b^{np}$. This rule helps us simplify expressions with multiple variables raised to certain powers.

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

A: To simplify an expression with a radical in the denominator, we need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. For example, if we have $\frac{1}{\sqrt{2}}$, we can rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{2}$.

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

A: A rational exponent is an exponent that can be expressed as a fraction, such as $\frac{1}{2}$ or $\frac{3}{4}$. An irrational exponent is an exponent that cannot be expressed as a fraction, such as $\sqrt{2}$ or $\pi$. Rational exponents can be simplified using the rule of exponents, while irrational exponents require a different approach.

Q: How do I simplify an expression with multiple variables raised to different powers?

A: To simplify an expression with multiple variables raised to different powers, we need to apply the rule of exponents separately to each variable. For example, if we have $\left(x^2 y^3\right)^4$, we can simplify it by applying the rule of exponents to each variable separately: $\left(x^2 y^3\right)^4 = x^{2 \times 4} y^{3 \times 4} = x^8 y^{12}$.

Q: What is the order of operations for simplifying expressions with exponents?

A: The order of operations for simplifying expressions with exponents is:

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponents.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.

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

A: To simplify an expression with a negative exponent, we need to rewrite the expression with a positive exponent. For example, if we have $x^{-2}$, we can rewrite it as $\frac{1}{x^2}$.

Q: What is the difference between a variable raised to a power and a variable multiplied by itself?

A: A variable raised to a power, such as $x^2$, represents the variable multiplied by itself a certain number of times. For example, $x^2$ represents $x \times x$. A variable multiplied by itself, such as $x \times x$, represents the variable raised to the power of 2.

Q: How do I simplify an expression with a variable raised to a fractional power?

A: To simplify an expression with a variable raised to a fractional power, we need to apply the rule of exponents. For example, if we have $x^{\frac{1}{2}}$, we can simplify it by rewriting it as $\sqrt{x}$.

Q: What is the relationship between exponents and roots?

A: Exponents and roots are related in that they represent the same mathematical operation. For example, $x^2$ represents the square of $x$, while $\sqrt{x}$ represents the square root of $x$. Similarly, $x^3$ represents the cube of $x$, while $\sqrt[3]{x}$ represents the cube root of $x$.

Q: How do I simplify an expression with multiple variables raised to different fractional powers?

A: To simplify an expression with multiple variables raised to different fractional powers, we need to apply the rule of exponents separately to each variable. For example, if we have $\left(x^{\frac{1}{2}} y^{\frac{3}{4}}\right)^2$, we can simplify it by applying the rule of exponents to each variable separately: $\left(x^{\frac{1}{2}} y^{\frac{3}{4}}\right)^2 = x^{\frac{1}{2} \times 2} y^{\frac{3}{4} \times 2} = x^1 y^{\frac{3}{2}}$.

Q: What is the difference between a variable raised to a power and a variable multiplied by itself a certain number of times?

A: A variable raised to a power, such as $x^2$, represents the variable multiplied by itself a certain number of times. For example, $x^2$ represents $x \times x$. A variable multiplied by itself a certain number of times, such as $x \times x \times x$, represents the variable raised to the power of 3.

Q: How do I simplify an expression with a variable raised to a negative fractional power?

A: To simplify an expression with a variable raised to a negative fractional power, we need to rewrite the expression with a positive fractional power. For example, if we have $x^{-\frac{1}{2}}$, we can rewrite it as $\frac{1}{x^{\frac{1}{2}}}$.

Q: What is the relationship between exponents and logarithms?

A: Exponents and logarithms are related in that they represent inverse operations. For example, $x^2$ represents the square of $x$, while $\log_x 2$ represents the logarithm of 2 to the base $x$. Similarly, $x^3$ represents the cube of $x$, while $\log_x 3$ represents the logarithm of 3 to the base $x$.

Q: How do I simplify an expression with multiple variables raised to different negative fractional powers?

A: To simplify an expression with multiple variables raised to different negative fractional powers, we need to rewrite each variable with a positive fractional power. For example, if we have $\left(x^{-\frac{1}{2}} y^{-\frac{3}{4}}\right)^2$, we can simplify it by rewriting each variable with a positive fractional power: $\left(x^{-\frac{1}{2}} y^{-\frac{3}{4}}\right)^2 = \left(\frac{1}{x^{\frac{1}{2}}} \frac{1}{y^{\frac{3}{4}}}\right)^2 = \frac{1}{x^{\frac{1}{2} \times 2} y^{\frac{3}{4} \times 2}} = \frac{1}{x^1 y^{\frac{3}{2}}}$.

Q: What is the difference between a variable raised to a power and a variable multiplied by itself a certain number of times?

A: A variable raised to a power, such as $x^2$, represents the variable multiplied by itself a certain number of times. For example, $x^2$ represents $x \times x$. A variable multiplied by itself a certain number of times, such as $x \times x \times x$, represents the variable raised to the power of 3.

Q: How do I simplify an expression with a variable raised to a fractional power and a variable multiplied by itself a certain number of times?

A: To simplify an expression with a variable raised to a fractional power and a variable multiplied by itself a certain number of times, we need to apply the rule of exponents and multiply the variables. For example, if we have $x^{\frac{1}{2}} \times x^2$, we can simplify it by applying the rule of exponents: $x^{\frac{1}{2}} \times x^2 = x^{\frac{1}{2} + 2} = x^{\frac{5}{2}}$.

Q: What is the relationship between exponents and roots?

A: Exponents and roots are related in that they represent the same mathematical operation. For example, $x^2$ represents the square of $x$, while $\sqrt{x}$ represents the square root of $x$. Similarly, $x^3$ represents the cube of $x$, while $\sqrt[3]{x}$ represents the cube root of $x$.

Q: How do I simplify an expression with multiple variables raised to different fractional powers and multiplied by each other?

A: To simplify an expression with multiple variables raised to different fractional powers and multiplied by each other, we need to apply the rule of exponents and multiply the variables. For example, if we have $x^{\frac{1}{2}} \times y^{\frac{3}{4}} \times z^2$, we can simplify it by applying the rule of exponents: $x^{\frac{1}{2}} \times y^{\frac{3}{4}} \times z^2 = x^{\frac{1}{2} + 0 + 2} \times y^{\frac{3}{4} + 0} = x^{\frac{5}{2}} \times y^{\frac{3}{4}}$.

Q: What is the difference between a variable raised to a power and a variable multiplied by itself a certain number of times?

A: A variable raised to a power, such as $x^2