Factorise 16a⁴-646⁴ ​

by ADMIN 22 views

Introduction

Factorising is a fundamental concept in algebra that allows us to simplify complex expressions by breaking them down into their constituent parts. In this article, we will explore the factorisation of the difference of two powers, specifically the expression 16a⁴ - 646⁴. We will use the difference of two squares and the difference of two cubes formulas to simplify this expression.

The Difference of Two Squares Formula

The difference of two squares formula states that:

a² - b² = (a + b)(a - b)

This formula can be used to factorise expressions of the form a² - b², where a and b are any real numbers.

The Difference of Two Cubes Formula

The difference of two cubes formula states that:

a³ - b³ = (a - b)(a² + ab + b²)

This formula can be used to factorise expressions of the form a³ - b³, where a and b are any real numbers.

Factorising 16a⁴ - 646⁴

To factorise the expression 16a⁴ - 646⁴, we can use the difference of two squares formula. We can rewrite the expression as:

(2a²)² - (26)²

Now, we can apply the difference of two squares formula:

(2a²)² - (26)² = (2a² + 26)(2a² - 26)

However, this is not the final factorisation. We can further simplify the expression by factoring out a common factor of 2 from the first term:

(2a² + 26)(2a² - 26) = 2(a² + 13)(a² - 13)

Now, we can see that the expression can be factored as:

2(a² + 13)(a² - 13)

Using the Difference of Two Cubes Formula

However, we can also use the difference of two cubes formula to factorise the expression 16a⁴ - 646⁴. We can rewrite the expression as:

(2a²)³ - (26)³

Now, we can apply the difference of two cubes formula:

(2a²)³ - (26)³ = (2a² - 26)((2a²)² + (2a²)(26) + (26)²)

Simplifying the expression, we get:

(2a² - 26)(4a⁴ + 52a² + 676)

However, this is not the final factorisation. We can further simplify the expression by factoring out a common factor of 2 from the first term:

2(a² - 13)(2a² + 13)(2a² + 26)

Now, we can see that the expression can be factored as:

2(a² - 13)(2a² + 13)(2a² + 26)

Conclusion

In this article, we have explored the factorisation of the difference of two powers, specifically the expression 16a⁴ - 646⁴. We have used the difference of two squares and the difference of two cubes formulas to simplify this expression. We have shown that the expression can be factored as:

2(a² - 13)(a² + 13)

or

2(a² - 13)(2a² + 13)(2a² + 26)

We hope that this article has provided a clear understanding of the factorisation of the difference of two powers and has shown how to use the difference of two squares and the difference of two cubes formulas to simplify complex expressions.

Common Factorisation Mistakes

When factorising expressions, it is easy to make mistakes. Here are some common mistakes to avoid:

  • Not checking for common factors: Make sure to check for common factors before applying the difference of two squares or the difference of two cubes formula.
  • Not simplifying the expression: Make sure to simplify the expression after applying the formula.
  • Not using the correct formula: Make sure to use the correct formula for the type of expression you are factorising.

Practice Problems

Here are some practice problems to help you practice factorising expressions:

  • Factorise the expression x² - 9.
  • Factorise the expression x³ - 27.
  • Factorise the expression 16x⁴ - 256.

Solutions

Here are the solutions to the practice problems:

  • x² - 9 = (x + 3)(x - 3)
  • x³ - 27 = (x - 3)(x² + 3x + 9)
  • 16x⁴ - 256 = 2(4x² - 16)(4x² + 16)

Conclusion

In this article, we have explored the factorisation of the difference of two powers, specifically the expression 16a⁴ - 646⁴. We have used the difference of two squares and the difference of two cubes formulas to simplify this expression. We have shown that the expression can be factored as:

2(a² - 13)(a² + 13)

or

2(a² - 13)(2a² + 13)(2a² + 26)

Introduction

In our previous article, we explored the factorisation of the difference of two powers, specifically the expression 16a⁴ - 646⁴. We used the difference of two squares and the difference of two cubes formulas to simplify this expression. In this article, we will answer some common questions related to factorising the difference of two powers.

Q: What is the difference of two squares formula?

A: The difference of two squares formula is:

a² - b² = (a + b)(a - b)

This formula can be used to factorise expressions of the form a² - b², where a and b are any real numbers.

Q: What is the difference of two cubes formula?

A: The difference of two cubes formula is:

a³ - b³ = (a - b)(a² + ab + b²)

This formula can be used to factorise expressions of the form a³ - b³, where a and b are any real numbers.

Q: How do I factorise an expression using the difference of two squares formula?

A: To factorise an expression using the difference of two squares formula, follow these steps:

  1. Check if the expression is in the form a² - b².
  2. If it is, then apply the formula: a² - b² = (a + b)(a - b).
  3. Simplify the expression by multiplying out the brackets.

Q: How do I factorise an expression using the difference of two cubes formula?

A: To factorise an expression using the difference of two cubes formula, follow these steps:

  1. Check if the expression is in the form a³ - b³.
  2. If it is, then apply the formula: a³ - b³ = (a - b)(a² + ab + b²).
  3. Simplify the expression by multiplying out the brackets.

Q: What are some common mistakes to avoid when factorising expressions?

A: Some common mistakes to avoid when factorising expressions include:

  • Not checking for common factors before applying the difference of two squares or the difference of two cubes formula.
  • Not simplifying the expression after applying the formula.
  • Not using the correct formula for the type of expression you are factorising.

Q: How do I check for common factors?

A: To check for common factors, follow these steps:

  1. Look for any common factors in the expression.
  2. If you find any, then factor them out of the expression.
  3. Simplify the expression by multiplying out the brackets.

Q: What are some examples of expressions that can be factorised using the difference of two squares formula?

A: Some examples of expressions that can be factorised using the difference of two squares formula include:

  • x² - 9
  • x² - 16
  • x² - 25

Q: What are some examples of expressions that can be factorised using the difference of two cubes formula?

A: Some examples of expressions that can be factorised using the difference of two cubes formula include:

  • x³ - 27
  • x³ - 64
  • x³ - 125

Conclusion

In this article, we have answered some common questions related to factorising the difference of two powers. We have provided examples of expressions that can be factorised using the difference of two squares and the difference of two cubes formulas. We hope that this article has provided a clear understanding of the factorisation of the difference of two powers and has shown how to use the difference of two squares and the difference of two cubes formulas to simplify complex expressions.

Practice Problems

Here are some practice problems to help you practice factorising expressions:

  • Factorise the expression x² - 49.
  • Factorise the expression x³ - 216.
  • Factorise the expression 16x⁴ - 256.

Solutions

Here are the solutions to the practice problems:

  • x² - 49 = (x + 7)(x - 7)
  • x³ - 216 = (x - 6)(x² + 6x + 36)
  • 16x⁴ - 256 = 2(4x² - 16)(4x² + 16)

Conclusion

In this article, we have provided a clear understanding of the factorisation of the difference of two powers and have shown how to use the difference of two squares and the difference of two cubes formulas to simplify complex expressions. We hope that this article has been helpful in providing a clear understanding of this important concept in algebra.