Which Expression Is Equivalent To $\sqrt[3]{x^5 Y}$?A. $x^{\frac{5}{3}} Y$ B. $x^{\frac{5}{3}} Y^{\frac{1}{3}}$ C. $x^{\frac{3}{5}} Y$ D. $x^{\frac{3}{5}} Y^3$

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Understanding the Problem

When dealing with radicals, it's essential to understand the properties of exponents and how they interact with the radical sign. In this problem, we're given the expression x5y3\sqrt[3]{x^5 y} and asked to determine which of the provided options is equivalent.

Properties of Radicals

To solve this problem, we need to recall the properties of radicals, specifically the property that states ann=a\sqrt[n]{a^n} = a. This property allows us to simplify radicals by removing the radical sign and reducing the exponent.

Applying the Properties of Radicals

Let's apply this property to the given expression x5y3\sqrt[3]{x^5 y}. We can rewrite the expression as x53⋅y3\sqrt[3]{x^5} \cdot \sqrt[3]{y}.

Simplifying the Expression

Now, we can simplify the expression by applying the property mentioned earlier. We have x53=x53\sqrt[3]{x^5} = x^{\frac{5}{3}} and y3=y13\sqrt[3]{y} = y^{\frac{1}{3}}.

Combining the Simplified Expressions

By combining the simplified expressions, we get x53⋅y13x^{\frac{5}{3}} \cdot y^{\frac{1}{3}}. This is the equivalent expression to x5y3\sqrt[3]{x^5 y}.

Comparing with the Options

Now, let's compare our result with the options provided:

A. x53yx^{\frac{5}{3}} y B. x53y13x^{\frac{5}{3}} y^{\frac{1}{3}} C. x35yx^{\frac{3}{5}} y D. x35y3x^{\frac{3}{5}} y^3

Conclusion

Based on our analysis, the correct answer is option B: x53y13x^{\frac{5}{3}} y^{\frac{1}{3}}. This is the only option that matches our simplified expression.

Final Thoughts

In conclusion, when dealing with radicals, it's essential to understand the properties of exponents and how they interact with the radical sign. By applying these properties, we can simplify radicals and determine which expression is equivalent to the given radical.

Frequently Asked Questions

Q: What is the property of radicals that allows us to simplify them?

A: The property of radicals that allows us to simplify them is ann=a\sqrt[n]{a^n} = a.

Q: How do we simplify the expression x5y3\sqrt[3]{x^5 y}?

A: We can simplify the expression by rewriting it as x53⋅y3\sqrt[3]{x^5} \cdot \sqrt[3]{y} and then applying the property mentioned earlier.

Q: What is the equivalent expression to x5y3\sqrt[3]{x^5 y}?

A: The equivalent expression to x5y3\sqrt[3]{x^5 y} is x53y13x^{\frac{5}{3}} y^{\frac{1}{3}}.

Additional Resources

For more information on radicals and exponents, check out the following resources:

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

A: A radical is a mathematical operation that involves finding the root of a number, while an exponent is a mathematical operation that involves raising a number to a power.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to find the largest perfect square that divides the radicand (the number inside the radical sign). You can then rewrite the radicand as the product of the perfect square and another number, and simplify the expression.

Q: What is the property of radicals that allows us to simplify them?

A: The property of radicals that allows us to simplify them is ann=a\sqrt[n]{a^n} = a.

Q: How do I multiply radicals?

A: To multiply radicals, you need to multiply the numbers inside the radical signs and then simplify the result.

Q: How do I divide radicals?

A: To divide radicals, you need to divide the numbers inside the radical signs and then simplify the result.

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, while an irrational exponent is an exponent that cannot be expressed as a fraction.

Q: How do I simplify a rational exponent?

A: To simplify a rational exponent, you need to rewrite the exponent as a product of a rational number and a power of the base.

Q: What is the property of exponents that allows us to simplify them?

A: The property of exponents that allows us to simplify them is am⋅an=am+na^m \cdot a^n = a^{m+n}.

Q: How do I add and subtract exponents?

A: To add and subtract exponents, you need to have the same base and the same exponent, and then add or subtract the exponents.

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

A: A positive exponent is an exponent that is greater than zero, while a negative exponent is an exponent that is less than zero.

Q: How do I simplify a negative exponent?

A: To simplify a negative exponent, you need to rewrite the exponent as a positive exponent and then simplify the result.

Q: What is the property of exponents that allows us to simplify negative exponents?

A: The property of exponents that allows us to simplify negative exponents is a−n=1ana^{-n} = \frac{1}{a^n}.

Q: How do I simplify a radical expression with a negative radicand?

A: To simplify a radical expression with a negative radicand, you need to rewrite the radicand as the product of a perfect square and another number, and then simplify the expression.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed as a rational or irrational number, while an imaginary number is a number that cannot be expressed as a rational or irrational number.

Q: How do I simplify an imaginary number?

A: To simplify an imaginary number, you need to rewrite the number as the product of a real number and an imaginary unit.

Q: What is the property of imaginary numbers that allows us to simplify them?

A: The property of imaginary numbers that allows us to simplify them is i2=−1i^2 = -1.

Q: How do I add and subtract imaginary numbers?

A: To add and subtract imaginary numbers, you need to have the same imaginary unit and then add or subtract the real parts.

Q: What is the difference between a complex number and a real number?

A: A complex number is a number that can be expressed as the sum of a real number and an imaginary number, while a real number is a number that can be expressed as a rational or irrational number.

Q: How do I simplify a complex number?

A: To simplify a complex number, you need to rewrite the number as the sum of a real number and an imaginary number, and then simplify the expression.

Q: What is the property of complex numbers that allows us to simplify them?

A: The property of complex numbers that allows us to simplify them is i2=−1i^2 = -1.

Q: How do I add and subtract complex numbers?

A: To add and subtract complex numbers, you need to have the same imaginary unit and then add or subtract the real parts.

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

A: A rational number is a number that can be expressed as a fraction, while an irrational number is a number that cannot be expressed as a fraction.

Q: How do I simplify a rational number?

A: To simplify a rational number, you need to rewrite the number as the product of a rational number and a power of the base.

Q: What is the property of rational numbers that allows us to simplify them?

A: The property of rational numbers that allows us to simplify them is ab=cd\frac{a}{b} = \frac{c}{d}.

Q: How do I add and subtract rational numbers?

A: To add and subtract rational numbers, you need to have the same denominator and then add or subtract the numerators.

Q: What is the difference between a rational number and a real number?

A: A rational number is a number that can be expressed as a fraction, while a real number is a number that can be expressed as a rational or irrational number.

Q: How do I simplify a real number?

A: To simplify a real number, you need to rewrite the number as the sum of a rational number and an irrational number, and then simplify the expression.

Q: What is the property of real numbers that allows us to simplify them?

A: The property of real numbers that allows us to simplify them is a2=a\sqrt{a^2} = a.

Q: How do I add and subtract real numbers?

A: To add and subtract real numbers, you need to have the same sign and then add or subtract the absolute values.

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed as a rational or irrational number, while a complex number is a number that can be expressed as the sum of a real number and an imaginary number.

Q: How do I simplify a complex number?

A: To simplify a complex number, you need to rewrite the number as the sum of a real number and an imaginary number, and then simplify the expression.

Q: What is the property of complex numbers that allows us to simplify them?

A: The property of complex numbers that allows us to simplify them is i2=−1i^2 = -1.

Q: How do I add and subtract complex numbers?

A: To add and subtract complex numbers, you need to have the same imaginary unit and then add or subtract the real parts.

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

A: A rational number is a number that can be expressed as a fraction, while an irrational number is a number that cannot be expressed as a fraction.

Q: How do I simplify a rational number?

A: To simplify a rational number, you need to rewrite the number as the product of a rational number and a power of the base.

Q: What is the property of rational numbers that allows us to simplify them?

A: The property of rational numbers that allows us to simplify them is ab=cd\frac{a}{b} = \frac{c}{d}.

Q: How do I add and subtract rational numbers?

A: To add and subtract rational numbers, you need to have the same denominator and then add or subtract the numerators.

Q: What is the difference between a rational number and a real number?

A: A rational number is a number that can be expressed as a fraction, while a real number is a number that can be expressed as a rational or irrational number.

Q: How do I simplify a real number?

A: To simplify a real number, you need to rewrite the number as the sum of a rational number and an irrational number, and then simplify the expression.

Q: What is the property of real numbers that allows us to simplify them?

A: The property of real numbers that allows us to simplify them is a2=a\sqrt{a^2} = a.

Q: How do I add and subtract real numbers?

A: To add and subtract real numbers, you need to have the same sign and then add or subtract the absolute values.

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed as a rational or irrational number, while a complex number is a number that can be expressed as the sum of a real number and an imaginary number.

Q: How do I simplify a complex number?

A: To simplify a complex number, you need to rewrite the number as the sum of a real number and an imaginary number, and then simplify the expression.

Q: What is the property of complex numbers that allows us to simplify them?

A: The property of complex numbers that allows us to simplify them is i2=−1i^2 = -1.

Q: How do I add and subtract complex numbers?

A: To add and subtract complex numbers, you need to have the same imaginary unit and then add or subtract the real parts.

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

A: A rational number is a number that can be expressed as a fraction, while an irrational number is a number that cannot be expressed as a fraction.

Q: How do I simplify a rational number?

A: To simplify a rational number, you need to rewrite the number