Chapter 8.7, Problem 59PFA

To determine

To find the expression equivalent to $x^{16}-y^{16}.$

Answer to Problem 59PFA

  $(x^8+y^8)(x^4+y^4)(x^2+y^2)(x-y)(x+y)$

Explanation of Solution

Given:

Expression: $x^{16}-y^{16}$

Concept used:

Factoring the expression using difference of squares.

Difference of squares: $a^2-b^2=(a+b)(a-b)$

Calculation:

Consider the given expression,

  $\Rightarrow x^{16}-y^{16}$

Applying the difference of squares,

  $\Rightarrow ({(x^8)}^2-{(y^8)}^2)=(x^8+y^8)(x^8-y^8)$

Further, applying the difference of squares,

  $\begin{array}{l}\Rightarrow (x^8-y^8)\\ $ \Rightarrow ({(x^4)}^2-{(y^4)}^2)$\Rightarrow (x^4+y^4)(x^4-y^4)\end{array}$

Further, applying the difference of squares,

  $\begin{array}{l}\Rightarrow (x^4-y^4)\\ $ \Rightarrow ({(x^2)}^2-{(y^2)}^2)$\Rightarrow (x^2-y^2)(x^2+y^2)\end{array}$

Further, again applying the difference of squares,

  $\begin{array}{l}\Rightarrow (x^2-y^2)\\ $ \Rightarrow ({(x)}^2-{(y)}^2)$\Rightarrow (x-y)(x+y)\end{array}$

Therefore,

  $\Rightarrow x^{16}-y^{16}=(x^8+y^8)(x^4+y^4)(x^2+y^2)(x-y)(x+y)$

Conclusion:

Therefore, the expression equivalent to the given expression is

  $(x^8+y^8)(x^4+y^4)(x^2+y^2)(x-y)(x+y).$