MHB Can You Simplify This Complex Algebraic Expression?

[mathdad]

Factor

(x^2 + 1)^(3/2) + (x^2 + 1)^(7/2)

Solution:

(x^2 + 1)^(3/2)[1 + (x^2 + 1)^(7/12)]

Correct?

 

[MarkFL]

When you factor out the expression $x^2+1$, you factor out the term having the smallest exponent (which you did), and then you subtract that exponent from the terms:

$$(x^2+1)^{\frac{3}{2}}+(x^2+1)^{\frac{7}{2}}=(x^2+1)^{\frac{3}{2}}\left((x^2+1)^{\frac{3}{2}-\frac{3}{2}}+(x^2+1)^{\frac{7}{2}-\frac{3}{2}}\right)=(x^2+1)^{\frac{3}{2}}\left((x^2+1)^{0}+(x^2+1)^{\frac{4}{2}}\right)=(x^2+1)^{\frac{3}{2}}\left(1+(x^2+1)^{2}\right)$$

 

[mathdad]

(x^2 + 1)^(3/2)(1 + (x^2 + 1)^2)

What about simplifying the right expression more?

Right Expression:

1 + (x^2 + 1)^2

1 + (x^2 + 1)(x^2 + 1)

1 + x^4 + 2x^2 + 1

x^4 + 2x^2 + 1

Final answer:

(x^2 + 1)^(3/2)(x^4 + 2x^2 + 1)

Correct?

 

[MarkFL]

(x^2 + 1)^(3/2)(1 + (x^2 + 1)^2)

What about simplifying the right expression more?

Right Expression:

1 + (x^2 + 1)^2

1 + (x^2 + 1)(x^2 + 1)

1 + x^4 + 2x^2 + 1

x^4 + 2x^2 + 1

You've dropped one of the 1's there...:D

 

[mathdad]

Answer: (x^2 + 1)^(3/2)(x^4 + 2x^2 + 2)