(5^4 )^1 /2

Question

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 

 

 

 

Answer 1

 ( 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 

 

 

 

   

Answer 2

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 √5^² 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√x^{a  =} x^a/b For your problem ²√5⁴  = 5^4/2 = 5²= 25