Simplify the following. ( 4 3 − 5 ) 2 (4\sqrt3-5)^2

Folks. So let's take a look at this next practice problem. Here we have something that looks a little bit different. We've got four radical three minus five, and the entire parentheses gets squared. So in other words, when we have this whole entire parentheses squared, another way to write this is it's really just four radical three minus five times four radical three minus five. Now we can see is that we have two terms that either have a plus or a minus sign, so we're going to have to use the foil method. So we have to go first by multiplying these two terms and then the outer terms. So these ones over here, then the inner two terms. So these ones over here, and then finally, the last two terms are these over here. Alright. So go ahead and pause the video if you haven't already done so, but try to actually work out the multiplication there. I'm going to go ahead and do this over here. The first two terms would just be four radical three times four radical three. Alright, so just multiply by it by itself. Now the outer two terms, what happens is I have a four radical three that's positive times a negative five. So I have to stick a minus sign in there. I have minus and then I've got four radical three times five. Right, so times five. Then for our inner two terms, we're going to have something that's very similar. Here we have a negative five times a four radical three. So in other words, it's kind of just the same thing. There's a minus sign. We're going to have a minus four radical three times five. Alright. And then finally, what we have is we have the negative five and the negative five. Those two negatives will cancel, so you're going to have to add a plus sign here. You're gonna get 25. Alright? So now let's simplify each of the terms individually. So what happens here is if I go four times four, that becomes 16. And then if I have radical three times radical three, we've seen before is that the square root will undo the fact that we're squaring, the two threes inside of the radicals. So in other words, radical three times radical three, you just drop the square root sign. This actually just becomes three over here. Now what happens is I've got negative four, radical three times five. You multiply the non radical parts. This is going to be negative 20 radical three and then over here the same exact thing happens. Negative four times five over here becomes negative 20 radical three. Now finally for this last step you for this 25 you don't do anything else and then finally what we do here is we just simplify these first two terms. 16 times three becomes 48 and then we have is negative 20 radical three and then another negative 20 radical three. Now, because we are adding and subtracting two radical expressions that have the same radicand, the negative radical three, we actually can combine those things together. If they're not the same, you can't, but in this case, you can. So negative 20 times another or sorry, added to another negative 20 becomes a negative 40. So let's write that red. The negative 40 radical three. Alright? So these two things can combine because they have the same radicand. That's something we've seen before, and then you're going to add a 25. So now we're looking through this expression over here. What we have is we have one whole number 48 and another whole number here, which is 25. So the last thing is we could do is we can combine those as well. This just becomes 48 plus 25. This is going to become 73 minus 40 radical three. So there are no variables in this expression, but with this radical, we can't we can't, we can't simplify this any further. Alright? So this would be the most simplified that you would write this. So again, when you have these types of expressions, you're gonna use the foil method, and then just work one by one using your rules and your different, product rules and your different distribution methods to go ahead and simplify the terms. Thanks for watching. I'll see you in the next

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Intermediate Algebra

9. Roots, Radicals, and Complex Numbers

Multiplying, Dividing, and Rationalizing Radicals

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Multiple Choice

Simplify the following.
$(4 \sqrt{3} - 5)^{2}$

$73 - 40 \sqrt{3}$

$- 9 - 20 \sqrt{3}$

$41 - 20 \sqrt{3}$

$23 - 40 \sqrt{3}$

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Verified step by step guidance

Recognize that the expression is a binomial squared: $(4\sqrt{3} - 5)^2$. Use the formula for the square of a binomial: $(a - b)^2 = a^2 - 2ab + b^2$.

Identify $a = 4\sqrt{3}$ and $b = 5$. Substitute these into the formula: $(4\sqrt{3})^2 - 2 \times (4\sqrt{3}) \times 5 + 5^2$.

Calculate each term separately: $(4\sqrt{3})^2 = 4^2 \times (\sqrt{3})^2 = 16 \times 3$, $-2 \times 4\sqrt{3} \times 5 = -40\sqrt{3}$, and \$5^2 = 25$.

Combine the simplified terms: $16 \times 3 - 40\sqrt{3} + 25$.

Add the like terms (the constants) to get the simplified expression in the form $A - B\sqrt{3}$.

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Struggling with Intermediate Algebra?

Simplify the following.(43−5)2(4\(\sqrt\)3-5)^2

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Simplify the following.
$(4 \sqrt{3} - 5)^{2}$