Question

Question #34bd6

Answer 1

`(3x)^4(9x^2 + 625)`

Can also be written as

`(3x)^4)*((3x)^2 + 5^4)`

Explanation:

`(-3x)^6 + (15x)^4`

`( (-3)^6 * x^6) + ((15)^4 * x^4)`

`(3^6 * x^6) + (3^4 * 5^4 * x^4) as (-3)^6 = 3^6`

Taking common terms out,
`= (3^4* x^4) * ((3^2 * x^2) + 5^4)`

`= (3x)^4 (9x^2 + 625)`

Answer 2

`81x^4(9x^2+625)`

Explanation:

First lets try a test using easy numbers to see if the numbers behave as hoped:

Consider the test
`(4x)^2 =16x^2-> (2x xx2)^2 = (2x)^2xx2^2 = 4x^2xx4=16x^2`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

Using the above approach lets 'force' common factors to occur.
Note that `(-3)xx(-5)=+15`

Given: `(-3x)^6+(15x)^4`

write as: `[-3x]^6+[-3x xx(-5)]^4`

This is the same as:

`color(white)("d")[(-3x)^4(-3x)^2] + [(-3x)^4(-5)^4]`

Factoring out `(-3x)^4[(-3x)^2+(-5)^4]`

Note that `(-3x)^n` where `n` is even gives a positive value
So `(-3x)^4=(+3x)^4`. Also `(-5)^4` is even.

`(3x)^4[9x^2+ 5^4]`

`81x^4(9x^2+625)`