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Express d times 4d to the fourth to the one half power as a radical expression.

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So the key thing to realise here is that if I take anything

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to the one half power by definition, this is the

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same thing as taking the principle root

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of that anything.

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So let's just apply that here we have

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4d to the fourth raised to the one half power.

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So this part, right over here, let me do it in two different colours,

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so I have d right over there,

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and the I have 4d to the fourth raised to the one half power

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which by definition is the principle root,

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of 4d to the fourth and we explore other videos

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while this all works out well with exponent properties.

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Now we know how to simplify this right here.

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We are looking for the principle root of

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this is the same thing, this is the same thing as d

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times the principle root of 4 times the principle root

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of d to the fourth, times the principle root of d to the fourth

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The principle root of four, principle root of four is

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2, and we have that d out front.

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And the principle root of d to the 4th power is the

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I should do it as a positive square root of d to the four

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That is d squared, and we don't have to put an absolute value here,

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because even if d is a negative value,

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d squared will still be a positive value.

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And if this was a d squared here,

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then we would have to think about the absolute value over here.

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And then, just to simplify this we have a d times

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a 2 times a d squared that's multiply the d's times each other

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same base, that become d to the third.

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So this becomes 2 times d to the third power.

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Now another way you could have done this is

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to just straight use the exponent properties.

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You could have said that that is the same thing as

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d times and inside the parenthesis

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If you take the product of things and then raise them

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to an exponent, that's the same thing as raising

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each of them individually to the exponent

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and then taking the product.

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So that's the same thing

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so I'm gonna rewrite this part right over here

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as 4 to the one half power

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times d to the fourth to the one half power

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and 4 to the one half power, this is literally just

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the principle square root of 4 which is 2 and d to the fourth

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to the one half power, this is, again

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you can use exponent property, if I have something

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raised to some power and then I have it raised to

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another power, this is the same thing as

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that same base raised to the product of these powers

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Let me prove it in other videos.

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So this right over here, this expression right over here

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is going to be equal to d to the 4 times one half power.

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What's 4 times one half?

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Well that's just 2! So this simplifies once again

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d times 2 times d squared, which is 2 d to the 3rd power.