# Why can't the square root of a^2 + b^2 be simplified?

May 31, 2015

If we substitute a and b to equal 6 for example
it would be $\sqrt{{6}^{2} + {6}^{2}}$ it would equal 8.5(1.d.p) as it would be written as $\sqrt{36 + 36}$ giving a standard form as $\sqrt{72}$

However if it was ${\sqrt{6}}^{2} + {\sqrt{6}}^{2}$ it would equal 12 as the sqrt and $^ 2$ would cancel out to give the equation 6+6

Therefore $\sqrt{{a}^{2} + {b}^{2}}$ cannot be simplified unless given a substitution for a and b.

I hope this isn't too confusing.

May 31, 2015

Suppose we try to find a 'simpler' expression than $\sqrt{{a}^{2} + {b}^{2}}$

Such an expression would have to involve square roots or $n$th roots or fractional exponents somewhere along the way.

Hayden's example of $\sqrt{{6}^{2} + {6}^{2}}$ shows this, but let's go simpler:

If $a = 1$ and $b = 1$ then $\sqrt{{a}^{2} + {b}^{2}} = \sqrt{2}$

$\sqrt{2}$ is irrational. (Easy, but slightly lengthy to prove, so I won't here)

So if putting $a$ and $b$ into our simpler expression only involved addition, subtraction, multiplication and/or division of terms with rational coefficients then we would not be able to produce $\sqrt{2}$.

Therefore any expression for $\sqrt{{a}^{2} + {b}^{2}}$ must involve something beyond addition, subtraction, multiplication and/or division of terms with rational coefficients. In my book that would be no simpler than the original expression.