Simplify the following. 5 x ( 4 + x ) \sqrt{5x}(4+\sqrt{x})

Folks. So let's get started with this next one over here. So we have radical five x outside the parentheses that gets multiplied into four plus radical x. We're going to simplify the following here. Again, there's a plus or a minus sign. We have a term that's outside, so we're going to distribute. This radical five distributes to the four and also to the radical x the second term, and we end up with two terms over here. Here what happens is we have a number like four that's going to be multiplied to a radical expression, so we just use the four first. We write four times radical five x. Now we can't really simplify that because we can't simplify the five or the x inside the radical. But the second term over here now becomes we have radical five x times radical x over here. Now, if you already have a feeling as to how this is going to work out, you can go ahead just head and skip to the end. But if not, we're going to work out the steps. So with this first term here, it can't simplify any further. We've got four times radical five x. And and again, we can't do anything there. So we have plus and that what happens is we have this five x that's in the first radical that's going to multiply with the radical x of the second. Under the product rule, these things all get together and under one radical, we could just write five x squared because this x and this x will multiply together. Now we've seen this before. What happens is we have a square root of five, nothing happens there, but then we have a square root of an x squared. That basically cancels out the squaring, and we could just bring out one factor of X to the outside of the radical. So now what happens here for this final step is we have four times radical five X plus we bring one factor of x to the outside. And then what happens is we have a square root of five over here. Alright. Now we look through these terms four radical five x can't be simplified any further. We're going to add another expression with a radical in which the radicals are not the same. We have x times radical five and this can't be simplified any further. Thanks for watching. Let's move on to the next one.

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Multiplying, Dividing, and Rationalizing Radicals

Beginning & Intermediate Algebra

11. Roots, Radicals, and Complex Numbers

Multiplying, Dividing, and Rationalizing Radicals

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Multiple Choice

Simplify the following.
$\sqrt{5 x} (4 + \sqrt{x})$

$4 \sqrt{5 x} + \sqrt{10 x}$

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

$10 \sqrt{x} + x \sqrt{5}$

$10 \sqrt{x} + \sqrt{10 x}$

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Verified step by step guidance

Start with the expression $\sqrt{5x}(4 + \sqrt{x})$. The goal is to simplify this by distributing the square root term over the sum inside the parentheses.

Apply the distributive property: multiply $\sqrt{5x}$ by each term inside the parentheses separately, giving $\sqrt{5x} \times 4 + \sqrt{5x} \times \sqrt{x}$.

Rewrite the multiplication with constants and square roots: $4\sqrt{5x} + \sqrt{5x} \cdot \sqrt{x}$.

Use the property of square roots that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ to combine the second term: $\sqrt{5x} \cdot \sqrt{x} = \sqrt{5x \cdot x} = \sqrt{5x^2}$.

Simplify $\sqrt{5x^2}$ by separating the perfect square: $\sqrt{5} \cdot \sqrt{x^2} = \sqrt{5} \cdot x$. So the expression becomes $4\sqrt{5x} + x\sqrt{5}$.

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Struggling with Beginning & Intermediate Algebra?

Simplify the following.5x(4+x)\(\sqrt{5x}\)(4+\(\sqrt{x}\))

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Simplify the following.
$\sqrt{5 x} (4 + \sqrt{x})$