Factorise Fully: $ B^{12} - 625 }$Answer ${ B^{12 - 625 = \square }$

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Introduction

In mathematics, factorisation is a fundamental concept that involves breaking down a complex expression into simpler components. It is a crucial skill that is used extensively in various branches of mathematics, including algebra, geometry, and calculus. In this article, we will focus on factorising the expression $b^{12} - 625$, which is a difference of two powers. We will use various techniques, including the difference of squares and the sum and difference of cubes, to factorise this expression fully.

Understanding the Expression

The given expression is $b^{12} - 625$. This is a difference of two powers, where the first term is $b^{12}$ and the second term is $625$. To factorise this expression, we need to identify any common factors or patterns that can be used to simplify it.

Difference of Squares

One of the most common techniques used to factorise expressions is the difference of squares. The difference of squares formula is:

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

We can use this formula to factorise the expression $b^{12} - 625$. However, we need to express $625$ as a power of a number. Since $625 = 5^4$, we can rewrite the expression as:

$b^{12} - 5^4$

Now, we can use the difference of squares formula to factorise this expression:

$b^{12} - 5^4 = (b^6 + 5^2)(b^6 - 5^2)$

Sum and Difference of Cubes

The sum and difference of cubes formula is:

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

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

We can use this formula to factorise the expression $b^6 - 5^2$. Since $b^6 - 5^2$ is a difference of squares, we can use the difference of squares formula to factorise it further:

$b^6 - 5^2 = (b^3 + 5)(b^3 - 5)$

Factorising the Expression Fully

Now that we have factorised the expression $b^6 - 5^2$, we can use the sum and difference of cubes formula to factorise it fully:

$b^6 - 5^2 = (b^3 + 5)(b^3 - 5)$

$b^3 - 5 = (b - \sqrt[3]{5})(b^2 + b\sqrt[3]{5} + 5)$

$b^3 + 5 = (b + \sqrt[3]{5})(b^2 - b\sqrt[3]{5} + 5)$

Conclusion

In this article, we factorised the expression $b^{12} - 625$ fully using various techniques, including the difference of squares and the sum and difference of cubes. We expressed $625$ as a power of a number and used the difference of squares formula to factorise the expression. We then used the sum and difference of cubes formula to factorise the expression further. The final factorisation of the expression is:

$b^{12} - 625 = (b^6 + 5^2)(b^6 - 5^2)$

$b^6 - 5^2 = (b^3 + 5)(b^3 - 5)$

$b^3 - 5 = (b - \sqrt[3]{5})(b^2 + b\sqrt[3]{5} + 5)$

$b^3 + 5 = (b + \sqrt[3]{5})(b^2 - b\sqrt[3]{5} + 5)$

This factorisation provides a complete and detailed breakdown of the expression $b^{12} - 625$, and it can be used to solve various mathematical problems and equations.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Geometry" by H.S.M. Coxeter

Further Reading

• [1] "Factorisation" by Wikipedia
• [2] "Difference of Squares" by Math Open Reference
• [3] "Sum and Difference of Cubes" by Math Open Reference
  

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Introduction

In our previous article, we factorised the expression $b^{12} - 625$ fully using various techniques, including the difference of squares and the sum and difference of cubes. In this article, we will answer some frequently asked questions related to factorising this expression.

Q1: What is the difference of squares formula?

A1: The difference of squares formula is:

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

This formula can be used to factorise expressions that are in the form of a difference of squares.

Q2: How do I factorise the expression $b^{12} - 625$?

A2: To factorise the expression $b^{12} - 625$, you can follow these steps:

1. Express $625$ as a power of a number. In this case, $625 = 5^4$.
2. Use the difference of squares formula to factorise the expression $b^{12} - 5^4$.
3. Factorise the expression $b^6 - 5^2$ using the difference of squares formula.
4. Use the sum and difference of cubes formula to factorise the expression $b^6 - 5^2$ further.

Q3: What is the sum and difference of cubes formula?

A3: The sum and difference of cubes formula is:

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

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

This formula can be used to factorise expressions that are in the form of a sum or difference of cubes.

Q4: How do I factorise the expression $b^6 - 5^2$?

A4: To factorise the expression $b^6 - 5^2$, you can follow these steps:

1. Use the difference of squares formula to factorise the expression $b^6 - 5^2$.
2. Factorise the expression $b^3 - 5$ using the sum and difference of cubes formula.
3. Factorise the expression $b^3 + 5$ using the sum and difference of cubes formula.

Q5: What is the final factorisation of the expression $b^{12} - 625$?

A5: The final factorisation of the expression $b^{12} - 625$ is:

$b^{12} - 625 = (b^6 + 5^2)(b^6 - 5^2)$

$b^6 - 5^2 = (b^3 + 5)(b^3 - 5)$

$b^3 - 5 = (b - \sqrt[3]{5})(b^2 + b\sqrt[3]{5} + 5)$

$b^3 + 5 = (b + \sqrt[3]{5})(b^2 - b\sqrt[3]{5} + 5)$

Q6: Can I use the sum and difference of cubes formula to factorise the expression $b^{12} - 625$?

A6: Yes, you can use the sum and difference of cubes formula to factorise the expression $b^{12} - 625$. However, you need to express $625$ as a power of a number and use the difference of squares formula to factorise the expression $b^6 - 5^2$ before using the sum and difference of cubes formula.

Q7: What are some common mistakes to avoid when factorising the expression $b^{12} - 625$?

A7: Some common mistakes to avoid when factorising the expression $b^{12} - 625$ include:

• Not expressing $625$ as a power of a number.
• Not using the difference of squares formula to factorise the expression $b^6 - 5^2$.
• Not using the sum and difference of cubes formula to factorise the expression $b^6 - 5^2$ further.
• Not simplifying the expression $b^3 - 5$ and $b^3 + 5$ using the sum and difference of cubes formula.

Conclusion

In this article, we answered some frequently asked questions related to factorising the expression $b^{12} - 625$. We provided step-by-step instructions on how to factorise this expression using the difference of squares and the sum and difference of cubes formula. We also highlighted some common mistakes to avoid when factorising this expression.