Step by Step Solution
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Simplify :
sqrt(54x9y7)
STEP
1
:
Simplify the Integer part of the SQRT
Factor 54 into its prime factors
54 = 2 • 33
To simplify a square root, we extract factors which are squares, i.e., factors that are raised to an even exponent.
Factors which will be extracted are :
9 = 32
Factors which will remain inside the root are :
6 = 2 • 3
To complete this part of the simplification we take the squre root of the factors which are to be extracted. We do this by dividing their exponents by 2 :
3 = 3
At the end of this step the partly simplified SQRT looks like this:
3 • sqrt (6x9y7)
STEP
2
:
Simplify the Variable part of the SQRT
Rules for simplifing variables which may be raised to a power:
(1) variables with no exponent stay inside the radical
(2) variables raised to power 1 or (-1) stay inside the radical
(3) variables raised to an even exponent: Half the exponent taken out, nothing remains inside the radical. examples:
(3.1) sqrt(x8)=x4
(3.2) sqrt(x-6)=x-3
(4) variables raised to an odd exponent which is >2 or <(-2) , examples:
(4.1) sqrt(x5)=x2•sqrt(x)
(4.2) sqrt(x-7)=x-3•sqrt(x-1)
Applying these rules to our case we find out that
SQRT(x9y7) = x4y3 • SQRT(xy)
Combine both simplifications
sqrt (54x9y7) =
3 x4y3 • sqrt(6xy)
Simplified Root :
3 x4y3 • sqrt(6xy)
