To simplify the expression, we first simplify the square root terms:
√( 21 + 8 √5 ) = √(16 + 5 + 8√5) = √(16 + 5) + √(8√5) = √21 + 2√5
Now, substitute this back into the original expression:
√21 + 2√5 - √5 + 1
Now, combine like terms:
√21 + 2√5 - √5 + 1 = √21 + √5 + 1
So, √( 21 + 8 √5 ) - √5 + 1 simplifies to √21 + √5 + 1.
To simplify the expression, we first simplify the square root terms:
√( 21 + 8 √5 ) = √(16 + 5 + 8√5) = √(16 + 5) + √(8√5) = √21 + 2√5
Now, substitute this back into the original expression:
√21 + 2√5 - √5 + 1
Now, combine like terms:
√21 + 2√5 - √5 + 1 = √21 + √5 + 1
So, √( 21 + 8 √5 ) - √5 + 1 simplifies to √21 + √5 + 1.