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## (5^4 )^1 /2

In this discussion, you will simplify and compare equivalent expressions written both in radical form and with rational (fractional) exponents.
using the following words

Principal Root

Product rule

Quotient rule

Reciprocal

nth root

# Comments

Hi Michele;
I can apply the term PRINCIPAL ROOT to such discussion, but nothing else.  Is that good enough?  Are these instructions for this equation only, or for a list of equations?
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# 2 Answers

( 5 ^4 ) ^ ( 1/2) = 5 ^ ( 4 * 1/2 ) = 5 ^2  =25            / Use of identity ( a ^m) ^ n = a ^ ( mn)

√(5 ^ 4 )= (√ (5^4 ) = √[( 5 ^2)^ 2] = 5 ^2 = 25       / Definition : n√(a ^n) = a

# Comments

Be careful. √(a^2) = |a|, whereas  (√(a))^2 = a.
Hello Michelle.  I can help with some of this.

Principal Root= the non-negative square root of a number.  Your problem is 5 raised to the fourth raised to the 1/2.  raising a number to the 1/2 is the same as taking the square root.  SO you are taking the square root of 5 raised to the fourth.  5 raised to the fourth is 625.  The square root of 625 is 25

Product rule=the product rule is a great way to break down numbers that don't actually have square roots.  For example, the square root of 8 is equal to the square root of 4 multiplied by the square root of 2 which would be equal to 2√2.  In this case, your product rule would give you √5² multiplied by the  √5² , which again gives you 25

Quotient rule=n√x multiplied by n√y = n√xy  So for your problem, 2√52 multiplied by 2√52 = 2√625

Reciprocal Rule =b√xa  = xa/b For your problem 2√5= 54/2 = 52= 25