If 3 is the index, and there is no radicand, and inside of that radical is another radical with ten being the radicand, how could the result be 6 as the index with ten being the radicand? I thought you multiply the outer index with the inner one. In that case you would get 3, not 6.
the inner radical has an understood index of two (square root). 3x2 makes an index of 6.
To understand how the result could be 6 as the index with ten being the radicand, we need to clarify the principles of dealing with nested radicals.
When a radical is placed inside another radical, it is called a nested radical. To evaluate such an expression, we can follow a step-by-step process:
1. Start by simplifying the nested radical first.
In this case, the nested radical has 10 as the radicand. Let's find the value of this inner radical.
√10 = √(2 * 5) = √2 * √5
So, we've simplified the inner radical to the product of two radicals (√2 and √5).
2. Next, rewrite the original expression by replacing the nested radical with its simplified form.
In this case, the original expression was √( ) with no radicand specified. Let's replace it with the simplified version of the inner radical (√2 * √5).
√(√10) = √(√2 * √5)
3. Now, we can simplify the outer radical.
Since the expression is √(√10), it means that the index of the outer radical is 2.
√(√10) = [(√10)^(1/2)]^(1/2)
Applying the exponent rule for radicals, we can multiply the exponents:
√(√10) = (√10)^(1/4)
4. Finally, evaluate the resulting expression.
To calculate (√10)^(1/4), we raise the radicand (10) to the power of the reciprocal of the index (1/4).
(√10)^(1/4) = 10^(1/8) ≈ 1.455
Therefore, the result of the expression √(√10) is approximately 1.455, not 6 as you mentioned.
In summary, evaluating nested radicals involves simplifying the inner radical first, then substituting that simplified form into the original expression, and finally simplifying the outer radical using the exponent rule for radicals.