How do you simplify radicals? And what is root 16 and root 32 simplified?I'm having trouble remembering how radicals work. Root 8 simplified is 2 root 2 right? How do you get to such an answer?...

How do you simplify radicals? And what is root 16 and root 32 simplified?

I'm having trouble remembering how radicals work.

Root 8 simplified is 2 root 2 right?

How do you get to such an answer?

What is root 16 and root 32 simplified?

Asked on by francy17

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embizze | High School Teacher | (Level 1) Educator Emeritus

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A radical expression contains a radical sign. The expression "underneath" the radical is said to be the radicand. Thus for the expression `sqrt(a)` we have `a` as the radicand.

There are three rules for simplifying square roots:

(1) There can be no perfect square factor (other than 1) of the radicand. Factor the expression under the radical -- if any factor is a perfect square then it can be removed.

Ex: `sqrt(8)=sqrt(4*2)=sqrt(4)sqrt(2)=2sqrt(2)` . Here we used the property `sqrt(ab)=sqrt(a)sqrt(b)` . Note that 4 is a perfect square -- namely `4=2^2` .

Ex: `sqrt(16)=4` by evaluation.

Ex: `sqrt(32)=sqrt(16*2)=sqrt(16)sqrt(2)=4sqrt(2)`

Ex: `sqrt(12)=sqrt(4*3)=sqrt(4)sqrt(3)=2sqrt(3)` . We could have factored 12 as 2x6, but neither 2 nor 6 is a perfect square.

Ex: `sqrt(14)=sqrt(14)` -- it is already simplified. 14 factors, but no factor is a perfect square.

Ex: `sqrt(72a^3b^5)=sqrt(36*2*a^2*a*b^4*b)=sqrt(36a^2b^4)sqrt(2ab)=6ab^2sqrt(2ab)`

Note that we factored so that the largest perfect square factor is removed -- 72 can be factored as 4*18 or 9*8, but 36 is the largest square factor of 72. Likewise, `b^2` is a factor of `b^5` , but `b^4` is the largest square factor of `b^5` .

(2) There can be no fractions under the radical. Thus `sqrt(a/b)` is not allowed. Here we can use the property `sqrt(a/b)=(sqrt(a))/(sqrt(b))` .

(3) Finally, we are not allowed to have a radical in the denominator. So `sqrt(a)/sqrt(b)` is not allowed. We can "fix" such an expression by multiplying by a fancy form of one; namely `sqrt(b)/sqrt(b)`

Ex: `1/sqrt(2)=1/sqrt(2)*sqrt(2)/sqrt(2)=sqrt(2)/2` . When multiplying fractions you multiply the numerators, and multiply the denominators. For the denominator, `sqrt(2)sqrt(2)=sqrt(4)=2` or you can realize that the definition of `sqrt(2)` is the number when multiplied by itself gives you 2, and then realize we just multiplied it by itself.

Ex: `sqrt(25)/sqrt(9)=5/3` Just by evaluating the roots.

Ex: `sqrt(25)/sqrt(15)` : here we can use the property `sqrt(a)/sqrt(b)=sqrt(a/b)` to get `sqrt(25)/sqrt(15)=sqrt(25/15)=sqrt(5/3)` . This makes the numbers smaller.

This is still not simplified as it violates rule (2):`sqrt(5/3)=sqrt(5)/sqrt(3)` . This violates rule (3) so:`sqrt(5)/sqrt(3)=sqrt(5)/sqrt(3)*sqrt(3)/sqrt(3)=sqrt(15)/3` which is simplified.

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These rules can be extended to higher roots. Rules (2) and (3) are the same; fixing a rule (3) violation is a bit more complicated. Also, rule (1) must be modified: there can be no perfect `n^(th)` powers under the radical. I.e. there can be no perfect cubes if you are taking a cube root, etc...

Another expansion is for radical expressions in the denominator; for example `2/(1+sqrt(3))` is not allowed. You fix this by multiplying by the conjugate, i.e `2/(1+sqrt(3))*(1-sqrt(3))/(1-sqrt(3))=(2-2sqrt(3))/(-2)=-1+sqrt(3)`

 

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