1.check whether root-16/root448 is a rational number or irrational number.
2.if root3=1.732 then find value of square root 3-1/root3+1
since √448 = 8√7, which is clearly irrational, then
-16/8√7 = -2/√7 is clearly irrational
You probably meant (√3-1)/(√3+1) , but in the outcome it would not make any difference as to irrational vs rational
(√3-1)/(√3+1)
= (√3-1)/(√3+1) * (√3-1)/(√3-1)
= (3 - 2√3 + 1)/(3-1)
= 2 - √3, which is still irrational
using your approximate value of √ = appr 1.732
(√3-1)/(√3+1) = appr 2 - 1.732
= appr .268
To check whether a number is rational or irrational, we need to determine if it can be expressed as a fraction or not.
1. To check whether root(16)/root(448) is a rational or irrational number:
- First, simplify the expression by evaluating the roots. root(16) is 4 since 4 squared equals 16, and root(448) can be simplified to 8(root(7)).
- Now we have 4 / (8(root(7))).
- Next, divide the numerator and denominator by their greatest common divisor (GCD) to simplify the fraction. GCD(4, 8) = 4, so divide both by 4.
- The expression simplifies to 1 / (2(root(7))).
- We see that the denominator contains the root of a non-perfect square (root(7)). Therefore, root(16)/root(448) is an irrational number.
2. To find the value of root(3) - 1 / root(3) + 1 using the given value of root(3) as 1.732:
- Substitute the value of root(3) into the expression: 1.732 - 1 / 1.732 + 1.
- Simplify the numerator: 1.732 - 1 = 0.732.
- Simplify the denominator: 1.732 + 1 = 2.732.
- The final expression is 0.732 / 2.732, which is a rational number.