Lesson 40: Simplifying Radical Expressions

New Concept

The Product Rule for Radicals: $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ and $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.

What’s next

Next, you'll apply this rule to simplify and combine radical expressions, including those with variables.

Given that $a$ and $b$ are real numbers and $n$ is an integer greater than 1,

$\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25}\sqrt{2} = 5\sqrt{2}$
$\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36}\sqrt{2} = 6\sqrt{2}$
$\sqrt[3]{128} = \sqrt[3]{64 \cdot 2} = \sqrt[3]{64}\sqrt[3]{2} = 4\sqrt[3]{2}$

This rule is your secret weapon! It lets you break a big, scary number under a root into a product of smaller, manageable ones. Your mission is to find a perfect square factor hiding inside, pull its root out to the front, and leave the non-perfect leftover part inside. This makes simplifying complex radicals so much easier.

When radical expressions have the same radicand and index, they are like radicals and can be combined. This is analogous to combining like terms, such as $2x + 3x = 5x$.

$4\sqrt{5} + 9\sqrt{5} = (4+9)\sqrt{5} = 13\sqrt{5}$
$12\sqrt{7} - 3\sqrt{7} - 2\sqrt{7} = (12-3-2)\sqrt{7} = 7\sqrt{7}$
$\sqrt{18} + \sqrt{50} = \sqrt{9 \cdot 2} + \sqrt{25 \cdot 2} = 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$

Think of radicals like items with the same name. You can combine '3 apples' and '5 apples' to get '8 apples'. In math, you can combine $3\sqrt{2}$ and $5\sqrt{2}$ to get $8\sqrt{2}$. If the number and type of root are different (like $\sqrt{2}$ and $\sqrt{3}$), they're not 'like' and can't be combined!

When simplifying variables under a radical, all variables represent non-negative real numbers. To simplify $\sqrt{x^n}$, the result is $x^{\frac{n}{2}}$.

$\sqrt{x^8} = \sqrt{(x^4)^2} = x^4$
$\sqrt{y^9} = \sqrt{y^8 \cdot y} = \sqrt{y^8}\sqrt{y} = y^4\sqrt{y}$
$\sqrt{20a^5b^2} = \sqrt{4 \cdot 5 \cdot a^4 \cdot a \cdot b^2} = 2a^2b\sqrt{5a}$

Variables under radicals follow a 'two-for-one' escape plan for square roots. For every pair of identical variables, one gets to leave the radical house. If there's an odd variable out, it stays inside. This is a fun way to remember you're just dividing the exponent by 2. For example, $x^6$ inside becomes $x^3$ outside.