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By Shanelle P pd.8
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1. Find the largest perfect square and simplify it.
This will divide evenly and there will be no decimals,
remainders, or fractions.
2. Write the number under the radical as a product of
the perfect square and your answer from dividing.
This will help you to simplify.
3. Reduce the radical which you have now created.
4. if the radical has a perfect square, that goes
outside the radical sign and the number that cannot
be reduced stays inside.
5. if it is raised to an index over 2, you find the root.
It can be the cubed root, fourth root etc. 2^3=64.
and √64 raised to the cubed root is 2.
If there is no index, it is 2.
Base- the number used as a factor. In 5^3, 5
is the base.
}  Exponent- the number that tells you how
many times the base is used as a factor. In
5^3, 3 is the exponent.
}  Radicand -is the expression under the radical
sign. In , 3 is the radicand.
}  Index- tells you how you should simplify the
base. For example, in
, 3 is
the index.
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Perfect Square
Perfect Cube
4, 9, 16, 25, 36, 49, 64,81,100
2
4
6
8
x , x , x , x ,
2 2 2 4
6 8
x y ,x y , 16x y , These powers are even.
8, 27, 64,
125,216,343,512,72
9,1000
x3, x6, x9, x12, ...
x 3y 3, x 3y 6,
27x6y9, ...
These powers are
multiples of 3
}  To simplify means
to find another
expression with the same value.
}  HINT
– A Simplified Radical
doesn’t have a fraction under it.
√72= √36x2= 6√2
}  The largest perfect square in 72 is 36.
36x2=72. When you find the square root of
36, you get 6 , because 6x6 is 36. Since 6 is
the actual square root, It goes outside the
radical sign. 2 is left inside the radical sign
because it cannot be simplified anymore.
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√16x^4=4x^2
}  This radical has a variable inside. We know
that the index is 2 because it isn’t expressed.
X^4 can be simplified to x^2 because x^4 is
a perfect square. 16 is also a perfect square,
which is reduced to 4. And that is how you
get 4x^2
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√75 = √25x3 = 5 √3
}  75 has the square factor of 25. 25 times 3 is
75. 25 unsquared is 5, which goes outside of
the radical sign but 3 is left inside because it
cannot be broken down anymore, 3 isn’t a
perfect square.
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√54= √9x6= 3 √6
}  The largest perfect square in 54 is 9. 9
unsquared is 3, 6 isn’t a perfect square so it
will be left inside of the radical sign and 3 will
be put on the outside.
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3 √320= 3 √64x5= 4 3√5
}  Here, we have the radical raised to the third
root. The largest cubed root in 320 is 64. 64
multiplied by 5 is 320. 4x4x4 or 4^3 is 64. 4
is moved to the outside and 5 stays inside
because it isn’t a perfect square and it cannot
be simplified anymore.
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√4500 = √100x45= √10x9x5= √10x3x5=
30√5
4,500 is a large number. The largest perfect
square in 4500 is 100. 100x45=4500 so both
numbers will be under the radical sign. 100
unsquared is 10 and 45 can be broken down.
The largest perfect square in 45 is 9. 9
unsquared is 3. 10 and 3 are multiplied because
they are both perfect squares. 30 goes outside of
the radical sign and 5 remains since it can no
longer be broken down
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In this problem, we have two different variables
and a number already outside of the radical. The
perfect square in 18 is 9. 9x2=18. We can them
split up the problem. We combine the 2 with the
a because they cannot be simplified. It becomes
2a. 2a stays inside the radical. B^4 is simplified
and that goes outside the radical. So 3ab^2 is
outside the radical. But wait there’s already a
three outside. Whenever there is a number
outside of the radical, you multiply it by whatever
you took out the radical sign. So you multiply
3x3ab^2. That is how you get 9ab^2 on the
outside of the radical and 2a inside the radial.
3 is the index
}  ^3 √48=3^ √8x6= √4x6=2 ^3√6
}  There is a number in the index so we have to
find the largest perfect cube. In this case it is
8. 2x2x2=8. 8x6=48. The 2 goes outside of
the radical and the 6 in left inside. Notice the
index remains there.
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√4x5/4= 2√5/4 = √5/4
In this problem, we have a fraction. First we are
going to focus on the numerator. √20 is 4x5.
4 is a perfect square. 2x2=4. The numerator
becomes 2√5/4. We know that 4 is divisible
by 2 so we can simplify. 2/2=1, 4/2=2. The
answer can be simplified to √5/4.
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8√50+3√2=
√25x2+3√2=8x5√2=40√2+3√2=43√2
}  Whenever a number is outside of the radical,
we must multiply it. 50 can be simplified to
25x2. 5 goes outside the radical so we
multiply it by the 8.3√2 can no longer be
simplified. So 40√2+3√2=43√2.
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If the numbers inside the radical are different,
you cannot combine them.
}  Ex- 22√2+34√4.
the radicals cannot be combined.
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Simplify
}  √4a^2b^4
}  √180
}  √300
1. 2ab^2.
}  6√5
}  10√3
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