Arithmetic With Square Roots

Links and useful resources

Concept summary and lesson

Square roots have a few rules of engagement that you need to understand. When you get them down, they'll let you solve problems more easily. Here's what you need to know:

Products (and quotients) of square roots

You can multiply square roots, by just multiplying what's in the square root:

$\sqrt{a} \sqrt{b} = \sqrt{a b}$

Naturally, you can use this in reverse, dividing out factors of expressions into their own square root terms. This is pretty handy for calculating square roots of long numbers:

What is $\sqrt{0.000169}$ ?

\begin{aligned} \sqrt{0.000169} & = \sqrt{0.000001} \sqrt{169} & Multiplication of square roots \\ = \sqrt{\frac{1}{1000000}} \sqrt{169} & Equivalent fraction \\ = \frac{1}{1000} \sqrt{169} & Square root of fraction is square root of parts \\ = \frac{1}{1000} 13 = \frac{13}{1000} = 0.013 & squre root of 169=13 \end{aligned}

Sums and differences of square roots

Unfortunately, we can't do the same things with sums and differences. That is, $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$

To see why, all we have to do is work with the equation a bit:

\begin{aligned} Assume \sqrt{a + b} = \sqrt{a} + \sqrt{b} : \\ \sqrt{a + b} & = \sqrt{a} + \sqrt{b} & Assumed \\ (\sqrt{a + b})^{2} & = (\sqrt{a} + \sqrt{b})^{2} & squaring property of equality \\ a + b & = (\sqrt{a} + \sqrt{b})^{2} & Definition of square root \\ = (\sqrt{a} + \sqrt{b}) (\sqrt{a} + \sqrt{b}) & squaring \\ = (\sqrt{a})^{2} + \sqrt{a} \sqrt{b} + \sqrt{b} \sqrt{a} + (\sqrt{b})^{2} & distributive property \\ = a + b + 2 \sqrt{a} \sqrt{b} & reorder terms \\ a + b & \neq a + b + 2 \sqrt{a} \sqrt{b} & contradiction, unless a,b=0 \end{aligned}

Guided practice

Homework

#hw (math) (gr7) 9.3 Arithmetic with Square Roots (p. 363) 2025-02-26