Which expression is a difference of squares? 9mx^(4)−49nx^(6) 100xx^(2)−10yx^(4) 32ax^(2)−81nx^(2) 16hx^(2)−21tx^(10)

Question

Which expression is a difference of squares?

9m
`x^(4)`−49n
`x^(6)`

100x
`x^(2)`−10y
`x^(4)`

32a
`x^(2)`−81n
`x^(2)`

16h
`x^(2)`−21t
`x^(10)`

Answer 1

If these are only integers, the answer is the first example.

However, each of these expressions can be represented as the difference of squares.

`9m^4-49n^6=3^2(m^2)^2-7^2(n^3)^2=(3m^2)^2-(7n^3)^2\\\\100x^2-10y^4=10^2x^2-(√(10))^2(y^2)^2=(10x)^2-(y^2√(10))^2\\\\32a^2-81n^2=(√(32))^2a^2-9^2n^2=(a√(32))^2-(9n)^2\\\\16h^2-21t^(10)=4^2h^2-(√(21))^2(t^5)^2=(4h)^2-(t^5√(21))^2`

Used:

`(a^n)^m=a^(nm)\\\\(ab)^n=a^nb^n\\\\(√(a))^2=a`