Drag The Factors To The Correct Locations On The Image. Not All Factors Will Be Used.What Is The Factored Form Of This Expression? 27 M 3 + 125 N 3 27m^3 + 125n^3 27 M 3 + 125 N 3 $\begin{array}{ll} 3m + 5n & 3m - 5n \ 9m^2 - 8mn + 5n^2 & 9m + 25n \ 9m^2 - 15mn +

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Introduction

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Factoring the sum of cubes is a fundamental concept in algebra that allows us to express a sum of cubes as a product of two binomials. In this article, we will explore the factored form of the expression $27m^3 + 125n^3$ using the sum of cubes formula.

The Sum of Cubes Formula

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The sum of cubes formula states that for any two numbers $a$ and $b$, the sum of their cubes can be factored as:

$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$

Applying the Sum of Cubes Formula

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To factor the expression $27m^3 + 125n^3$, we need to identify the values of $a$ and $b$. In this case, we can let $a = 3m$ and $b = 5n$. Substituting these values into the sum of cubes formula, we get:

$$(3m)^3 + (5n)^3 = (3m + 5n)((3m)^2 - (3m)(5n) + (5n)^2)$$

Simplifying the Expression

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Simplifying the expression further, we get:

$$(3m + 5n)(9m^2 - 15mn + 25n^2)$$

Factored Form of the Expression

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Therefore, the factored form of the expression $27m^3 + 125n^3$ is:

$$(3m + 5n)(9m^2 - 15mn + 25n^2)$$

Conclusion

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In this article, we have explored the factored form of the expression $27m^3 + 125n^3$ using the sum of cubes formula. By identifying the values of $a$ and $b$ and applying the formula, we were able to simplify the expression and obtain its factored form.

Discussion

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The factored form of the expression $27m^3 + 125n^3$ can be used to solve equations and inequalities involving the sum of cubes. For example, if we want to find the values of $m$ and $n$ that satisfy the equation $27m^3 + 125n^3 = 0$, we can use the factored form to rewrite the equation as:

$$(3m + 5n)(9m^2 - 15mn + 25n^2) = 0$$

This equation can be solved by setting each factor equal to zero and solving for $m$ and $n$.

Example

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Let's consider an example to illustrate the use of the factored form of the expression $27m^3 + 125n^3$. Suppose we want to find the values of $m$ and $n$ that satisfy the equation $27m^3 + 125n^3 = 100$. We can use the factored form to rewrite the equation as:

$$(3m + 5n)(9m^2 - 15mn + 25n^2) = 100$$

To solve this equation, we can set each factor equal to zero and solve for $m$ and $n$. This will give us the values of $m$ and $n$ that satisfy the equation.

Tips and Tricks

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When factoring the sum of cubes, it's essential to identify the values of $a$ and $b$ correctly. In this case, we let $a = 3m$ and $b = 5n$. This allowed us to apply the sum of cubes formula and simplify the expression.

Common Mistakes

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When factoring the sum of cubes, it's common to make mistakes when identifying the values of $a$ and $b$. To avoid this, make sure to read the problem carefully and identify the values of $a$ and $b$ correctly.

Conclusion

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In conclusion, the factored form of the expression $27m^3 + 125n^3$ is $(3m + 5n)(9m^2 - 15mn + 25n^2)$. This can be used to solve equations and inequalities involving the sum of cubes. By following the steps outlined in this article, you can master the art of factoring the sum of cubes and solve a wide range of problems involving this concept.

Final Thoughts

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Factoring the sum of cubes is a powerful tool in algebra that allows us to express a sum of cubes as a product of two binomials. By mastering this concept, you can solve a wide range of problems involving the sum of cubes and become a proficient algebraist.

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Introduction

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In our previous article, we explored the factored form of the expression $27m^3 + 125n^3$ using the sum of cubes formula. In this article, we will answer some frequently asked questions about factoring the sum of cubes.

Q: What is the sum of cubes formula?

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A: The sum of cubes formula is a mathematical formula that allows us to express a sum of cubes as a product of two binomials. The formula is:

$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$

Q: How do I apply the sum of cubes formula?

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A: To apply the sum of cubes formula, you need to identify the values of $a$ and $b$. In the case of the expression $27m^3 + 125n^3$, we can let $a = 3m$ and $b = 5n$. Substituting these values into the formula, we get:

$$(3m)^3 + (5n)^3 = (3m + 5n)((3m)^2 - (3m)(5n) + (5n)^2)$$

Q: What is the factored form of the expression $27m^3 + 125n^3$?

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A: The factored form of the expression $27m^3 + 125n^3$ is:

$$(3m + 5n)(9m^2 - 15mn + 25n^2)$$

Q: How do I use the factored form to solve equations and inequalities?

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A: To use the factored form to solve equations and inequalities, you can set each factor equal to zero and solve for $m$ and $n$. For example, if we want to find the values of $m$ and $n$ that satisfy the equation $27m^3 + 125n^3 = 0$, we can use the factored form to rewrite the equation as:

$$(3m + 5n)(9m^2 - 15mn + 25n^2) = 0$$

Q: What are some common mistakes to avoid when factoring the sum of cubes?

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A: Some common mistakes to avoid when factoring the sum of cubes include:

• Not identifying the values of $a$ and $b$ correctly
• Not applying the sum of cubes formula correctly
• Not simplifying the expression correctly

Q: How can I practice factoring the sum of cubes?

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A: You can practice factoring the sum of cubes by working through examples and exercises. You can also try factoring different expressions and see if you can identify the values of $a$ and $b$ correctly.

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

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A: Factoring the sum of cubes has many real-world applications, including:

• Solving equations and inequalities in algebra
• Finding the roots of a polynomial equation
• Factoring quadratic expressions

Conclusion

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In conclusion, factoring the sum of cubes is a powerful tool in algebra that allows us to express a sum of cubes as a product of two binomials. By mastering this concept, you can solve a wide range of problems involving the sum of cubes and become a proficient algebraist.

Final Thoughts

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Factoring the sum of cubes is a fundamental concept in algebra that has many real-world applications. By practicing and mastering this concept, you can become a proficient algebraist and solve a wide range of problems involving the sum of cubes.

Additional Resources

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If you want to learn more about factoring the sum of cubes, here are some additional resources:

• Khan Academy: Factoring the Sum of Cubes
• Mathway: Factoring the Sum of Cubes
• Wolfram Alpha: Factoring the Sum of Cubes

Practice Problems

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Here are some practice problems to help you master factoring the sum of cubes:

• Factor the expression $64m^3 + 27n^3$
• Factor the expression $27m^3 + 125n^3$
• Factor the expression $125m^3 + 27n^3$

Conclusion

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In conclusion, factoring the sum of cubes is a powerful tool in algebra that allows us to express a sum of cubes as a product of two binomials. By mastering this concept, you can solve a wide range of problems involving the sum of cubes and become a proficient algebraist.