simplify each expression. write your answer with positive exponents assume that all variables represent positive real numbers.find the decimal value of each expression round approximate answer to four decimal places
(a^^1/2b)^1/2(ab^1/2)
multiply exponents when raising to a power:
(a^1/2b)^1/2 = a^1/4 b^1/2
add exponents when multiplying factors:
a^1/4 b^1/2 * a b^1/2 = a^5/4 b
To simplify the expression (a^(1/2)b)^(1/2)(ab^(1/2)), we can use the property of exponents that states (a^m)^n = a^(m*n).
First, let's simplify the expression step by step:
Step 1:
(a^(1/2)b)^(1/2) = a^((1/2)*(1/2)) * b^((1/2)*(1/2)) = a^(1/4) * b^(1/4)
Step 2:
(a^(1/4) * b^(1/4))(ab^(1/2)) = a^(1/4) * ab^(1/2) * b^(1/4)
Step 3:
Since a^(1/4) * a^(1/2) = a^((1/4) + (1/2)) = a^(3/4), we can simplify the expression further:
a^(1/4) * ab^(1/2) * b^(1/4) = a^(3/4) * b^(3/4)
Now, to find the decimal value of the expression, we need the specific values of 'a' and 'b'. Let's assume 'a' = 2 and 'b' = 3.
Substituting these values into the expression, we get:
(2^(3/4)) * (3^(3/4))
Calculating this expression using a calculator, we find that the decimal value is approximately 5.6382 (rounded to four decimal places).
Therefore, the decimal value of the expression (a^(1/2)b)^(1/2)(ab^(1/2)) is 5.6382 (rounded to four decimal places).