True or False: 9 + 16 \sqrt{9+16} 9 + 16 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> and 9 + 16 \sqrt9+\sqrt{16} 9 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> + 16 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> are equal.

Alright. So let's take a look at this problem here. So we're asked sort of a true or false question. Is the square root of 9 plus 16 all under one radical and square root of 9 plus square root of 16 different radicals, are those things equal to each other? Well, let's find out. If you have a square roots of 9 +16, that's all under the same radical. What ends up happening here is that you could basically treat this radical as like a parenthesis. You can do everything that's inside of the parenthesis or inside of the radical, and you can simplify before you take the square roots. So in other words, this really just becomes the square root of 25. We know that the square root of 25, the positive root, however, or is just is just 5. So that's what the square root of 9 +16 altogether is. So is that the same thing as square root of 9+thesquare root of 16? Well, what happens is square root of 9 +2square root of 16 is basically like saying 3+4, right? That's a perfect square, that's a perfect square, and this is actually just equal to 7. So notice how these two things are not equal to each other. Alright, so 1 is 5 and 1 is 7. So this is really really important. Again, this is why you can't actually just take 2 square roots and just sort of mash them together it's because that doesn't actually equal the same thing. This is equal to 5, but this expression is equal to 7. So it's actually false. So, this is false. Make sure that you don't do this. This is a big mistake, and you're gonna get the wrong answer. Alright? Thanks for watching.

0. Review of Algebra

Simplifying Radical Expressions

College Algebra

0. Review of Algebra

Simplifying Radical Expressions

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Multiple Choice

True or False:
$\sqrt{9+16}$ and $\sqrt9+\sqrt{16}$ are equal.

True

False

Cannot be determined

Verified step by step guidance

First, evaluate the expression inside the square root for the first term: $ 9 + 16 \sqrt{9 + 16} $. Calculate $ 9 + 16 $ to get $ 25 $.

Next, take the square root of the result from step 1: $ \sqrt{25} $, which simplifies to $ 5 $.

Now, multiply $ 16 $ by the result from step 2: $ 16 \times 5 = 80 $.

Add the result from step 3 to $ 9 $: $ 9 + 80 = 89 $. This is the value of the first expression.

For the second expression, $ 9 + 16 \sqrt{9} + \sqrt{16} $, calculate $ \sqrt{9} = 3 $ and $ \sqrt{16} = 4 $. Then, multiply $ 16 \times 3 = 48 $ and add $ 9 + 48 + 4 = 61 $. Compare the results: 89 and 61 are not equal, so the statement is False.