What Is ( X 2 Y 3 ) 3 X 2 Y 3 \frac{\left(x^2 Y^3\right)^3}{\sqrt[3]{x^2 Y}} 3 X 2 Y ​ ( X 2 Y 3 ) 3 ​ In Exponential Form?A. X 2 3 Y X 2 3 Y 1 3 \frac{x^{\frac{2}{3}} Y}{x^{\frac{2}{3}} Y^{\frac{1}{3}}} X 3 2 ​ Y 3 1 ​ X 3 2 ​ Y ​ B. X 7 3 Y 10 2 X 7 2 Y 4 3 \frac{x^{\frac{7}{3}} Y^{\frac{10}{2}}}{x^{\frac{7}{2}} Y^{\frac{4}{3}}} X 2 7 ​ Y 3 4 ​ X 3 7 ​ Y 2 10 ​ ​ C.

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Understanding Exponents and Radicals

Exponents and radicals are fundamental concepts in mathematics that help us simplify complex expressions and solve equations. In this article, we will explore the concept of exponents and radicals, and how to apply them to simplify the given expression $\frac{\left(x^2 y^3\right)^3}{\sqrt[3]{x^2 y}}$.

The Power of Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $x^2$ means $x \times x$, and $x^3$ means $x \times x \times x$. Exponents can be added, subtracted, multiplied, and divided, just like regular numbers.

Radicals and Roots

Radicals are a way of expressing the square root or other roots of a number. For example, $\sqrt{x}$ means the square root of $x$, and $\sqrt[3]{x}$ means the cube root of $x$. Radicals can be simplified by using the properties of exponents.

Simplifying the Expression

To simplify the expression $\frac{\left(x^2 y^3\right)^3}{\sqrt[3]{x^2 y}}$, we need to apply the properties of exponents and radicals.

Step 1: Simplify the Numerator

The numerator is $\left(x^2 y^3\right)^3$. Using the property of exponents that $(a^m)^n = a^{m \times n}$, we can simplify the numerator as follows:

$$\left(x^2 y^3\right)^3 = x^{2 \times 3} y^{3 \times 3} = x^6 y^9$$

Step 2: Simplify the Denominator

The denominator is $\sqrt[3]{x^2 y}$. Using the property of radicals that $\sqrt[n]{a} = a^{\frac{1}{n}}$, we can simplify the denominator as follows:

$$\sqrt[3]{x^2 y} = x^{\frac{2}{3}} y^{\frac{1}{3}}$$

Step 3: Simplify the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression as follows:

$$\frac{\left(x^2 y^3\right)^3}{\sqrt[3]{x^2 y}} = \frac{x^6 y^9}{x^{\frac{2}{3}} y^{\frac{1}{3}}}$$

Using the property of exponents that $\frac{a^m}{a^n} = a^{m - n}$, we can simplify the expression further as follows:

$$\frac{x^6 y^9}{x^{\frac{2}{3}} y^{\frac{1}{3}}} = x^{6 - \frac{2}{3}} y^{9 - \frac{1}{3}} = x^{\frac{16}{3}} y^{\frac{26}{3}}$$

Step 4: Simplify the Expression Further

We can simplify the expression further by combining the exponents of $x$ and $y$:

$$x^{\frac{16}{3}} y^{\frac{26}{3}} = \left(x^{\frac{16}{3}} y^{\frac{26}{3}}\right)$$

Step 5: Write the Expression in Exponential Form

The expression $\left(x^{\frac{16}{3}} y^{\frac{26}{3}}\right)$ is already in exponential form.

Conclusion

In this article, we have simplified the expression $\frac{\left(x^2 y^3\right)^3}{\sqrt[3]{x^2 y}}$ using the properties of exponents and radicals. We have shown that the expression can be simplified to $\left(x^{\frac{16}{3}} y^{\frac{26}{3}}\right)$, which is already in exponential form.

Answer

The correct answer is:

• $\boxed{\left(x^{\frac{16}{3}} y^{\frac{26}{3}}\right)}$

Discussion

This problem requires a good understanding of exponents and radicals. The student should be able to apply the properties of exponents and radicals to simplify the expression. The student should also be able to write the expression in exponential form.

Tips and Tricks

• Make sure to apply the properties of exponents and radicals correctly.
• Use the correct notation for exponents and radicals.
• Simplify the expression step by step.
• Check your answer by plugging in values for $x$ and $y$.

Practice Problems

• Simplify the expression $\frac{\left(x^3 y^2\right)^2}{\sqrt{x^3 y}}$.
• Simplify the expression $\frac{\left(x^4 y^3\right)^3}{\sqrt[3]{x^4 y}}$.
• Simplify the expression $\frac{\left(x^2 y^4\right)^2}{\sqrt{x^2 y}}$.

References

• Exponents and Radicals
• Properties of Exponents
• Radicals and Roots
  Q&A: Exponents and Radicals =============================

Frequently Asked Questions

In this article, we will answer some frequently asked questions about exponents and radicals.

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

A: An exponent is a shorthand way of writing repeated multiplication, while a radical is a way of expressing the square root or other roots of a number.

Q: How do I simplify an expression with exponents and radicals?

A: To simplify an expression with exponents and radicals, you need to apply the properties of exponents and radicals. This includes using the rules for multiplying and dividing exponents, as well as simplifying radicals.

Q: What is the rule for multiplying exponents?

A: The rule for multiplying exponents is that when you multiply two numbers with the same base, you add their exponents. For example, $x^2 \times x^3 = x^{2+3} = x^5$.

Q: What is the rule for dividing exponents?

A: The rule for dividing exponents is that when you divide two numbers with the same base, you subtract their exponents. For example, $x^5 \div x^3 = x^{5-3} = x^2$.

Q: How do I simplify a radical?

A: To simplify a radical, you need to find the largest perfect square that divides the number inside the radical. You can then take the square root of this perfect square and simplify the radical.

Q: What is the difference between a square root and a cube root?

A: A square root is the inverse operation of squaring a number, while a cube root is the inverse operation of cubing a number.

Q: How do I simplify an expression with a square root and a cube root?

A: To simplify an expression with a square root and a cube root, you need to apply the properties of radicals. This includes using the rules for multiplying and dividing radicals, as well as simplifying the expression.

Q: What is the rule for multiplying radicals?

A: The rule for multiplying radicals is that when you multiply two numbers with the same base, you multiply their exponents. For example, $\sqrt{x} \times \sqrt{y} = \sqrt{xy}$.

Q: What is the rule for dividing radicals?

A: The rule for dividing radicals is that when you divide two numbers with the same base, you subtract their exponents. For example, $\sqrt{x} \div \sqrt{y} = \sqrt{\frac{x}{y}}$.

Q: How do I simplify an expression with a square root and a cube root?

A: To simplify an expression with a square root and a cube root, you need to apply the properties of radicals. This includes using the rules for multiplying and dividing radicals, as well as simplifying the expression.

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 an expression with a rational exponent and an irrational exponent?

A: To simplify an expression with a rational exponent and an irrational exponent, you need to apply the properties of exponents. This includes using the rules for multiplying and dividing exponents, as well as simplifying the expression.

Conclusion

In this article, we have answered some frequently asked questions about exponents and radicals. We have covered the rules for multiplying and dividing exponents, as well as simplifying radicals and expressions with square roots and cube roots.

Practice Problems

• Simplify the expression $\frac{\left(x^3 y^2\right)^2}{\sqrt{x^3 y}}$.
• Simplify the expression $\frac{\left(x^4 y^3\right)^3}{\sqrt[3]{x^4 y}}$.
• Simplify the expression $\frac{\left(x^2 y^4\right)^2}{\sqrt{x^2 y}}$.

References

• Exponents and Radicals
• Properties of Exponents
• Radicals and Roots