What Is $125 X^9+64 Y^{12}$ Written As A Sum Of Cubes?A. $\left(25 X^3\right)^3+\left(4 Y^4\right)^3$B. $\left(5 X^3\right)^3+\left(16 Y^4\right)^3$C. $\left(25 X^3\right)^3+\left(8 Y^4\right)^3$D. $\left(5

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What is $125 x^9+64 y^{12}$ written as a sum of cubes?

Understanding the Concept of Sum of Cubes

A sum of cubes is an algebraic expression that can be written in the form a3+b3a^3 + b^3, where aa and bb are algebraic expressions. This concept is crucial in mathematics, particularly in algebra and number theory. In this article, we will explore how to write the given expression $125 x^9+64 y^{12}$ as a sum of cubes.

The Formula for Sum of Cubes

The formula for sum of cubes is given by:

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

This formula can be used to factorize any expression that can be written in the form a3+b3a^3 + b^3. However, in this case, we are given an expression that is not in the form a3+b3a^3 + b^3. We need to manipulate the given expression to fit the formula.

Manipulating the Given Expression

The given expression is $125 x^9+64 y^{12}$. We can rewrite this expression as:

125x9+64y12=(5x3)3+(4y4)3125 x^9+64 y^{12} = (5x^3)^3 + (4y^4)^3

This is because 125=53125 = 5^3 and 64=4364 = 4^3. Therefore, we can rewrite the given expression as a sum of cubes.

Evaluating the Options

Now that we have rewritten the given expression as a sum of cubes, we can evaluate the options.

A. (25x3)3+(4y4)3\left(25 x^3\right)^3+\left(4 y^4\right)^3

B. (5x3)3+(16y4)3\left(5 x^3\right)^3+\left(16 y^4\right)^3

C. (25x3)3+(8y4)3\left(25 x^3\right)^3+\left(8 y^4\right)^3

D. (5x3)3+(4y4)3\left(5 x^3\right)^3+\left(4 y^4\right)^3

We can see that option D is the correct answer, as it matches the expression we derived earlier.

Conclusion

In conclusion, the given expression $125 x^9+64 y^{12}$ can be written as a sum of cubes as (5x3)3+(4y4)3\left(5 x^3\right)^3+\left(4 y^4\right)^3. This is achieved by manipulating the given expression to fit the formula for sum of cubes. We can use this concept to factorize any expression that can be written in the form a3+b3a^3 + b^3.

Key Takeaways

  • The formula for sum of cubes is given by a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2).
  • The given expression $125 x^9+64 y^{12}$ can be rewritten as a sum of cubes as (5x3)3+(4y4)3\left(5 x^3\right)^3+\left(4 y^4\right)^3.
  • We can use the concept of sum of cubes to factorize any expression that can be written in the form a3+b3a^3 + b^3.

Further Reading

If you want to learn more about the concept of sum of cubes, I recommend checking out the following resources:

  • Khan Academy: Sum of Cubes
  • Mathway: Sum of Cubes
  • Wolfram MathWorld: Sum of Cubes

These resources provide a comprehensive overview of the concept of sum of cubes and how to apply it to factorize expressions.
Frequently Asked Questions (FAQs) about Sum of Cubes

Q: What is a sum of cubes?

A: A sum of cubes is an algebraic expression that can be written in the form a3+b3a^3 + b^3, where aa and bb are algebraic expressions.

Q: How do I identify a sum of cubes?

A: To identify a sum of cubes, look for expressions that can be written in the form a3+b3a^3 + b^3. For example, 8x3+27y38x^3 + 27y^3 is a sum of cubes because it can be written as (2x)3+(3y)3(2x)^3 + (3y)^3.

Q: What is the formula for sum of cubes?

A: The formula for sum of cubes is given by:

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Q: How do I factor a sum of cubes?

A: To factor a sum of cubes, use the formula:

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

For example, to factor 8x3+27y38x^3 + 27y^3, we can use the formula to get:

8x3+27y3=(2x+3y)((2x)2−(2x)(3y)+(3y)2)8x^3 + 27y^3 = (2x + 3y)((2x)^2 - (2x)(3y) + (3y)^2)

Q: Can I use the formula for sum of cubes to factor any expression?

A: No, the formula for sum of cubes can only be used to factor expressions that can be written in the form a3+b3a^3 + b^3. If the expression is not in this form, you will need to use a different method to factor it.

Q: What are some common mistakes to avoid when working with sum of cubes?

A: Some common mistakes to avoid when working with sum of cubes include:

  • Not recognizing that an expression is a sum of cubes
  • Not using the correct formula to factor the expression
  • Not simplifying the expression after factoring

Q: How do I simplify a factored sum of cubes?

A: To simplify a factored sum of cubes, look for any common factors in the expression. For example, if we have:

(2x+3y)((2x)2−(2x)(3y)+(3y)2)(2x + 3y)((2x)^2 - (2x)(3y) + (3y)^2)

We can simplify this expression by factoring out a common factor of 2x2x from the first term and a common factor of 3y3y from the second term.

Q: Can I use the formula for sum of cubes to solve equations?

A: Yes, the formula for sum of cubes can be used to solve equations that involve sum of cubes. For example, if we have the equation:

8x3+27y3=1258x^3 + 27y^3 = 125

We can use the formula to factor the left-hand side of the equation and then solve for xx and yy.

Q: What are some real-world applications of sum of cubes?

A: Sum of cubes has many real-world applications, including:

  • Algebraic geometry
  • Number theory
  • Cryptography
  • Computer science

These applications involve using the formula for sum of cubes to factor expressions and solve equations.

Q: How can I practice working with sum of cubes?

A: You can practice working with sum of cubes by:

  • Solving problems that involve sum of cubes
  • Factoring expressions that are sums of cubes
  • Simplifying factored sums of cubes
  • Using the formula for sum of cubes to solve equations

You can find many practice problems and resources online, including worksheets, videos, and interactive tools.