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In $3-38$ write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero.$$\sqrt{162 a^{4} b^{3}}+3-a b \sqrt{18 a^{2} b}-1$$

$6 a^{2} b \sqrt{2 b}+2$

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Chapter 3

REAL NUMBERS AND RADICALS

Section 4

Adding and Subtracting Radicals

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

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So in this exercise, we're gonna be simplifying the expression radical. 162 8 of the fourth beat of the third plus three minus a be radical 18 a squared B minus one. Okay, so the first thing I'm gonna do is try to simplify this radical as much as I can, and it's probably gonna match up with this radical in some way. So the first thing I recognize is I can rewrite this as two times 81 81 is a perfect square. I can factor that out. Can we write it in the fourth as a squared times a squared and I can rewrite beat of the third as b squared. That was a bad squared B squared times beat. So if I factor all these perfect square terms out, I'm gonna factor this 81 out and get a nine. I'm gonna fact this a squared out in this other a scored out. So that's eight times a So it's another a squared. I'm gonna factor this b squared ounce that's gonna be be out front in here. And I am left with a two in the radical and A B in their article. Okay, so let's see if we can do the same thing to this radical. Obviously, 18 can be re written as two times 99 is a perfect square. A squared is already a perfect square, so I can factor that out in front as well. So this radical becomes a B times three times another A. And I have a to be left in the Radicals. This whole expression is going to be three a squared B radical to be. And so and then obviously these unit terms right here can be combined separately. So three minus one is going to be too. So we're just gonna put that over there. For now. We have nine a squared B minus three a squared, be on the right, audible to bees. That's gonna be nine minus three. So that simplifies to six a squared B square root to be and then we still have a plus. Two on the outsides are final. Answer is going to be six a squared, be radical to be plus two

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