Add or subtract as indicated and simplify. 12 + 20 − 45 \sqrt{12}+\sqrt{20}-\sqrt{45}

Here we're asked to add or subtract as indicated and simplify. We're given here the square root of 12 plus the square root of 20 minus the square root of 45. Now these are currently not like radicals. They all have the same index, but they don't have the same radicand. But let's see if we can make them into like radicals so that we can do this addition and subtraction. Using the product rule here, I'm gonna break this square root of 12 into the square root of four times the square root of three. Then with the square root of 20, I'm going to go ahead and break this down into the square root of four times the square root of five. Again, here using the product rule because four times five is equal to 20. Then finally, with this 45, also breaking this up using the product rule, I'm gonna rewrite this as the square root of nine times the square root of five. Here, I can start to see what I think will be like radicals emerge. So let's do some simplification here. The square root of four is two, so I can go ahead and rewrite this root four times root three as two root three. So what was originally the square root of 12 is now two root three. Then adding that next term here, it was originally the square root of 20. Now I have the square root of four times five, which I can rewrite as two root five because, again, the square root of four is two. Then with that last term, it started off as the square root of of 45. We are subtracting here. And the square root of nine times the square root of five, I can rewrite this as three root five because the square root of nine is three. So now I do have two like radicals. Here I have two root five and three root five. These have the same index of two, they're both square roots, and the same radicand of five. So I can go ahead and actually subtract this. Two root five minus three root five will give me a negative one root five. I don't have to write that one. And I've combined those like radicals. Now I have that two, root three, we can't do any further combining here so this is my fully simplified answer. Let me know if you have any questions and I'll see you in the next one.

11. Roots, Radicals, and Complex Numbers

Adding and Subtracting Radicals

Beginning & Intermediate Algebra

11. Roots, Radicals, and Complex Numbers

Adding and Subtracting Radicals

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Multiple Choice

Add or subtract as indicated and simplify.
$\sqrt{12} + \sqrt{20} - \sqrt{45}$

$2 \sqrt{3} - \sqrt{5}$

$\sqrt{3} - 2 \sqrt{5}$

$4 \sqrt{3} + \sqrt{5}$

$4 \sqrt{5} + \sqrt{3}$

Verified step by step guidance

Start by expressing each square root in terms of its prime factors to simplify them. For example, write $\sqrt{12}$ as $\sqrt{4 \times 3}$, $\sqrt{20}$ as $\sqrt{4 \times 5}$, and $\sqrt{45}$ as $\sqrt{9 \times 5}$.

Use the property of square roots that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ to separate the factors inside each square root. This allows you to take out perfect squares from under the root.

Simplify each term by taking the square root of the perfect squares: $\sqrt{4} = 2$ and $\sqrt{9} = 3$. Rewrite each term as a product of an integer and a simplified square root.

Rewrite the original expression using the simplified terms: it will look like a sum and difference of terms with coefficients multiplied by square roots of numbers that cannot be simplified further.

Combine like terms by adding or subtracting the coefficients of the square roots that have the same radicand (the number inside the square root). This will give you the simplified expression.

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Add or subtract as indicated and simplify.12+20−45\sqrt{12}+\sqrt{20}-\sqrt{45}

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Add or subtract as indicated and simplify.
$\sqrt{12} + \sqrt{20} - \sqrt{45}$