Multiply Square Roots: Finding √10 × √3

Great question! When you multiply square roots, the result stays under the radical sign. $\sqrt{10} \cdot \sqrt{3} = \sqrt{30}$, not 30. The answer 30 would mean you're claiming $\sqrt{30} = 30$, which is false!

Yes, but only when multiplying! The rule $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ works for multiplication. For addition like $\sqrt{10} + \sqrt{3}$, you cannot combine them this way.

Check if 30 has any perfect square factors! Since 30 = 2 × 3 × 5 (all prime factors appear once), there are no perfect squares to factor out. So $\sqrt{30}$ is already in simplest form.

Square roots are fractional exponents! $\sqrt{10} = 10^{1/2}$ and $\sqrt{3} = 3^{1/2}$. Using the rule $x^n \cdot y^n = (xy)^n$, we get $10^{1/2} \cdot 3^{1/2} = (10 \cdot 3)^{1/2} = 30^{1/2} = \sqrt{30}$.

The same rule applies! For example, $\sqrt{50} \cdot \sqrt{8} = \sqrt{50 \cdot 8} = \sqrt{400} = 20$. Sometimes the result simplifies to a whole number, but often it stays as a radical.