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Let's discuss Simplified Radical Form. In
order for an algebraic expression to be
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simplified radical form, all of the
following must be true. [SOUND]The first
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property that must be hold, is that no
radican contains a factor, to a power of
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greater than or equal to the index of the
radical. For example, the cube root, of
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y^5, would not be considered simplified.
So this is not. Simplified, because the
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power of the factor y, namely five, is
greater than the index of the radical,
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which is three.
The second property that must hold is that
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no power of the radicand and the index of
the radical have a common factor other
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than one.
For example, the ninth root of x to the
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twelfth would not be simplified. Because
nine and twelve Have a common factor of
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three.
The third property that needs to hold, is
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that no radical appears in the
denominator. For example, two / seven.
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Is not simplified, because we have this
square root of seven in the denominator.
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And the last property that must hold is
that no fraction appears within a radical.
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For example, the square root of five /
four is not simplified. Because we have
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this 5/4, within the radical. Alright,
let's see an example of how we do put an
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algebraic into simplified radical form.
Let's put this expression into simplified
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radical form. And we're assuming here that
x and y represent positive real numbers.
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The first thing we should notice here is
that the radicand contains factors raised
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to powers greater than the index of four.
We have the six, as well as the nine.
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Since we are simplifying a fourth root, we
need to focus on the perfect fourth power
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factors of the radican, this 16X to the 6y
to the ninth.
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Now something is a perfect fourth power
factor when its exponent is a multiple of
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four.
So what do we have,? We have the fourth
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root of 16x to the six, y to the nine = to
the fourth root of sixteen.
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But sixteen is two to the fourth power.
And now we're going to extract the perfect
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fourth power factors here. So we're going
to rewrite x to the sixth as x to the
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fourth times x^2 and we're going to write
y to the ninth as y to the eighth times y.
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And this is equal to the fourt h root of
two^4x^4x^2 and then we are going to
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rewrite y^8 as (y^2)^4.
And then times y. Now, let's group
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together all perfect fourth power factors,
namely, two^4, x^4, and y^2^4.
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So this is equal to the fourth root of two
to the fourth, x to the fourth, and then y
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squared to the fourth, and then times x
squared y. And now by properties of
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exponents, this is equal to the fourth
root of 2xy^2, all raised to the fourth
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power. And then times x^2 y. Again by
properties of exponents this is equal to
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the fourth root of 2xy squared to the
fourth and then times the fourth root of x
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squared y. And now, this is equal to the
fourth root of 2xy^2^4 is 2xy^2 and then
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we still have this fourth root of x^2*y.
So the question is, are we done? And we're*
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not because two and four have a common
factor other than one.
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So let's first split this up as 2xy^2.
And then, the fourth root of x^2.
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And then the fourth root of y. Now, let's
convert this term here to rational
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exponent form. In other words, this is x.
x^2/4 which is x^1/2 or square root of x.
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Therefore this is equal to 2xy^2 times the
square root of x Times the fourth root of
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y, which would be in simplified radical
form. All right, and this is how we put an
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algebraic expression into simplified
radical form. Thank you, and we'll see you
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next time.