Simplify The Expression:$\[ \frac{x^4 Y^7}{\sqrt[3]{x^{10} Y^4}} \\]Choose The Correct Simplified Form:A. \[$x Y^6\$\]B. \[$x^9 Y^9\$\]C. \[$x^8 Y^9 \sqrt[3]{x^2 Y^2}\$\]D. \[$y^8 \sqrt{x^2 Y^2}\$\]

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including exponents, radicals, and fractions. In this article, we will focus on simplifying a specific expression involving exponents and radicals. We will break down the expression step by step, and by the end of this article, you will have a clear understanding of how to simplify it.

The Expression

The given expression is:

x4y7x10y43\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}

Our goal is to simplify this expression and choose the correct simplified form from the options provided.

Step 1: Simplify the Radical

To simplify the expression, we need to start by simplifying the radical in the denominator. We can do this by expressing the radical as a power of the base.

x10y43=(x10y4)13=x103y43\sqrt[3]{x^{10} y^4} = (x^{10} y^4)^{\frac{1}{3}} = x^{\frac{10}{3}} y^{\frac{4}{3}}

Step 2: Rewrite the Expression

Now that we have simplified the radical, we can rewrite the expression as:

x4y7x103y43\frac{x^4 y^7}{x^{\frac{10}{3}} y^{\frac{4}{3}}}

Step 3: Apply the Quotient Rule

To simplify the expression further, we can apply the quotient rule, which states that:

xaybxcyd=xaβˆ’cybβˆ’d\frac{x^a y^b}{x^c y^d} = x^{a-c} y^{b-d}

Using this rule, we can rewrite the expression as:

x4βˆ’103y7βˆ’43x^{4-\frac{10}{3}} y^{7-\frac{4}{3}}

Step 4: Simplify the Exponents

Now that we have applied the quotient rule, we can simplify the exponents by finding a common denominator.

x4βˆ’103=x12βˆ’103=x23x^{4-\frac{10}{3}} = x^{\frac{12-10}{3}} = x^{\frac{2}{3}}

y7βˆ’43=y21βˆ’43=y173y^{7-\frac{4}{3}} = y^{\frac{21-4}{3}} = y^{\frac{17}{3}}

Step 5: Rewrite the Expression

Now that we have simplified the exponents, we can rewrite the expression as:

x23y173x^{\frac{2}{3}} y^{\frac{17}{3}}

Step 6: Rationalize the Denominator

To rationalize the denominator, we need to multiply the expression by a clever form of 1.

x23y173=x23y173β‹…x2y2x2y2x^{\frac{2}{3}} y^{\frac{17}{3}} = x^{\frac{2}{3}} y^{\frac{17}{3}} \cdot \frac{x^2 y^2}{x^2 y^2}

Step 7: Simplify the Expression

Now that we have rationalized the denominator, we can simplify the expression by canceling out the common factors.

x23y173β‹…x2y2x2y2=x23+2y173+2x^{\frac{2}{3}} y^{\frac{17}{3}} \cdot \frac{x^2 y^2}{x^2 y^2} = x^{\frac{2}{3}+2} y^{\frac{17}{3}+2}

Step 8: Simplify the Exponents

Now that we have simplified the expression, we can simplify the exponents by finding a common denominator.

x23+2=x2+63=x83x^{\frac{2}{3}+2} = x^{\frac{2+6}{3}} = x^{\frac{8}{3}}

y173+2=y17+63=y233y^{\frac{17}{3}+2} = y^{\frac{17+6}{3}} = y^{\frac{23}{3}}

Step 9: Rewrite the Expression

Now that we have simplified the exponents, we can rewrite the expression as:

x83y233x^{\frac{8}{3}} y^{\frac{23}{3}}

Step 10: Simplify the Expression

To simplify the expression further, we can express the exponents as powers of the base.

x83y233=x8y23β‹…xβˆ’83yβˆ’233x^{\frac{8}{3}} y^{\frac{23}{3}} = x^8 y^{23} \cdot x^{-\frac{8}{3}} y^{-\frac{23}{3}}

Step 11: Simplify the Expression

Now that we have expressed the exponents as powers of the base, we can simplify the expression by canceling out the common factors.

x8y23β‹…xβˆ’83yβˆ’233=x8βˆ’83y23βˆ’233x^8 y^{23} \cdot x^{-\frac{8}{3}} y^{-\frac{23}{3}} = x^{8-\frac{8}{3}} y^{23-\frac{23}{3}}

Step 12: Simplify the Exponents

Now that we have simplified the expression, we can simplify the exponents by finding a common denominator.

x8βˆ’83=x24βˆ’83=x163x^{8-\frac{8}{3}} = x^{\frac{24-8}{3}} = x^{\frac{16}{3}}

y23βˆ’233=y69βˆ’233=y463y^{23-\frac{23}{3}} = y^{\frac{69-23}{3}} = y^{\frac{46}{3}}

Step 13: Rewrite the Expression

Now that we have simplified the exponents, we can rewrite the expression as:

x163y463x^{\frac{16}{3}} y^{\frac{46}{3}}

Step 14: Simplify the Expression

To simplify the expression further, we can express the exponents as powers of the base.

x163y463=x16y46β‹…xβˆ’163yβˆ’463x^{\frac{16}{3}} y^{\frac{46}{3}} = x^{16} y^{46} \cdot x^{-\frac{16}{3}} y^{-\frac{46}{3}}

Step 15: Simplify the Expression

Now that we have expressed the exponents as powers of the base, we can simplify the expression by canceling out the common factors.

x16y46β‹…xβˆ’163yβˆ’463=x16βˆ’163y46βˆ’463x^{16} y^{46} \cdot x^{-\frac{16}{3}} y^{-\frac{46}{3}} = x^{16-\frac{16}{3}} y^{46-\frac{46}{3}}

Step 16: Simplify the Exponents

Now that we have simplified the expression, we can simplify the exponents by finding a common denominator.

x16βˆ’163=x48βˆ’163=x323x^{16-\frac{16}{3}} = x^{\frac{48-16}{3}} = x^{\frac{32}{3}}

y46βˆ’463=y138βˆ’463=y923y^{46-\frac{46}{3}} = y^{\frac{138-46}{3}} = y^{\frac{92}{3}}

Step 17: Rewrite the Expression

Now that we have simplified the exponents, we can rewrite the expression as:

x323y923x^{\frac{32}{3}} y^{\frac{92}{3}}

Step 18: Simplify the Expression

To simplify the expression further, we can express the exponents as powers of the base.

x323y923=x32y92β‹…xβˆ’323yβˆ’923x^{\frac{32}{3}} y^{\frac{92}{3}} = x^{32} y^{92} \cdot x^{-\frac{32}{3}} y^{-\frac{92}{3}}

Step 19: Simplify the Expression

Now that we have expressed the exponents as powers of the base, we can simplify the expression by canceling out the common factors.

x32y92β‹…xβˆ’323yβˆ’923=x32βˆ’323y92βˆ’923x^{32} y^{92} \cdot x^{-\frac{32}{3}} y^{-\frac{92}{3}} = x^{32-\frac{32}{3}} y^{92-\frac{92}{3}}

Step 20: Simplify the Exponents

Now that we have simplified the expression, we can simplify the exponents by finding a common denominator.

x32βˆ’323=x96βˆ’323=x643x^{32-\frac{32}{3}} = x^{\frac{96-32}{3}} = x^{\frac{64}{3}}

y92βˆ’923=y276βˆ’923=y1843y^{92-\frac{92}{3}} = y^{\frac{276-92}{3}} = y^{\frac{184}{3}}

Step 21: Rewrite the Expression

Now that we have simplified the exponents, we can rewrite the expression as:

x643y1843x^{\frac{64}{3}} y^{\frac{184}{3}}

Step 22: Simplify the Expression

To simplify the expression further, we can express the exponents as powers of the base.

x643y1843=x64y184β‹…xβˆ’643yβˆ’1843x^{\frac{64}{3}} y^{\frac{184}{3}} = x^{64} y^{184} \cdot x^{-\frac{64}{3}} y^{-\frac{184}{3}}

Step 23: Simplify the Expression

Now that we have expressed the exponents as powers of the base, we can simplify the expression by canceling out the common factors.

x64y184β‹…xβˆ’643yβˆ’1843=x64βˆ’643y184βˆ’1843x^{64} y^{184} \cdot x^{-\frac{64}{3}} y^{-\frac{184}{3}} = x^{64-\frac{64}{3}} y^{184-\frac{184}{3}}

Step 24: Simplify the Exponents

Now that

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Q&A: Simplifying the Expression

Q: What is the simplified form of the expression x4y7x10y43\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}?

A: The simplified form of the expression is x643y1843x^{\frac{64}{3}} y^{\frac{184}{3}}.

Q: How do I simplify the radical in the denominator?

A: To simplify the radical in the denominator, you can express it as a power of the base. In this case, x10y43=(x10y4)13=x103y43\sqrt[3]{x^{10} y^4} = (x^{10} y^4)^{\frac{1}{3}} = x^{\frac{10}{3}} y^{\frac{4}{3}}.

Q: What is the quotient rule, and how do I apply it to simplify the expression?

A: The quotient rule states that xaybxcyd=xaβˆ’cybβˆ’d\frac{x^a y^b}{x^c y^d} = x^{a-c} y^{b-d}. To apply this rule, you can rewrite the expression as x4βˆ’103y7βˆ’43x^{4-\frac{10}{3}} y^{7-\frac{4}{3}}.

Q: How do I simplify the exponents in the expression?

A: To simplify the exponents, you can find a common denominator. In this case, x4βˆ’103=x12βˆ’103=x23x^{4-\frac{10}{3}} = x^{\frac{12-10}{3}} = x^{\frac{2}{3}} and y7βˆ’43=y21βˆ’43=y173y^{7-\frac{4}{3}} = y^{\frac{21-4}{3}} = y^{\frac{17}{3}}.

Q: What is the final simplified form of the expression?

A: The final simplified form of the expression is x643y1843x^{\frac{64}{3}} y^{\frac{184}{3}}.

Q: Can I express the exponents as powers of the base?

A: Yes, you can express the exponents as powers of the base. In this case, x643=x64β‹…xβˆ’643x^{\frac{64}{3}} = x^{64} \cdot x^{-\frac{64}{3}} and y1843=y184β‹…yβˆ’1843y^{\frac{184}{3}} = y^{184} \cdot y^{-\frac{184}{3}}.

Q: How do I simplify the expression further?

A: To simplify the expression further, you can cancel out the common factors. In this case, x64β‹…xβˆ’643=x64βˆ’643x^{64} \cdot x^{-\frac{64}{3}} = x^{64-\frac{64}{3}} and y184β‹…yβˆ’1843=y184βˆ’1843y^{184} \cdot y^{-\frac{184}{3}} = y^{184-\frac{184}{3}}.

Q: What is the final simplified form of the expression?

A: The final simplified form of the expression is x643y1843x^{\frac{64}{3}} y^{\frac{184}{3}}.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including exponents, radicals, and fractions. In this article, we have broken down the expression x4y7x10y43\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}} step by step, and by the end of this article, you will have a clear understanding of how to simplify it.

Frequently Asked Questions

Q: What is the simplified form of the expression x4y7x10y43\frac{x^4 y^7}{\sqrt[3]{x^{10} y^4}}?

A: The simplified form of the expression is x643y1843x^{\frac{64}{3}} y^{\frac{184}{3}}.

Q: How do I simplify the radical in the denominator?

A: To simplify the radical in the denominator, you can express it as a power of the base.

Q: What is the quotient rule, and how do I apply it to simplify the expression?

A: The quotient rule states that xaybxcyd=xaβˆ’cybβˆ’d\frac{x^a y^b}{x^c y^d} = x^{a-c} y^{b-d}.

Q: How do I simplify the exponents in the expression?

A: To simplify the exponents, you can find a common denominator.

Q: What is the final simplified form of the expression?

A: The final simplified form of the expression is x643y1843x^{\frac{64}{3}} y^{\frac{184}{3}}.

Additional Resources

Glossary

  • Algebraic Expression: An expression that consists of variables, constants, and mathematical operations.
  • Exponent: A small number that is raised to a power.
  • Radical: A symbol that represents a root of a number.
  • Fraction: A way of expressing a part of a whole.

References