If $\sqrt{2916} = 54$ then what is the value of the following? $\sqrt{29.16} + \sqrt{0.2916} + \sqrt{0.002916} + \sqrt{0.00002916}$
The problem asks for the value of the expression $\sqrt{29.16} + \sqrt{0.2916} + \sqrt{0.002916} + \sqrt{0.00002916}$, given that $\sqrt{2916} = 54$. We can evaluate each term by relating it back to the given value.
Evaluating Square Root Terms
We are given $\sqrt{2916} = 54$. We can use this to find the square roots of numbers with decimals:
- $\sqrt{29.16}$: This can be written as $\sqrt{2916 / 100}$. Using the property $\sqrt{a/b} = \sqrt{a} / \sqrt{b}$, we get: $\sqrt{29.16} = \sqrt{2916} / \sqrt{100} = 54 / 10 = 5.4$
- $\sqrt{0.2916}$: This can be written as $\sqrt{2916 / 10000}$. $\sqrt{0.2916} = \sqrt{2916} / \sqrt{10000} = 54 / 100 = 0.54$
- $\sqrt{0.002916}$: This can be written as $\sqrt{2916 / 1000000}$. $\sqrt{0.002916} = \sqrt{2916} / \sqrt{1000000} = 54 / 1000 = 0.054$
- $\sqrt{0.00002916}$: This can be written as $\sqrt{2916 / 100000000}$. $\sqrt{0.00002916} = \sqrt{2916} / \sqrt{100000000} = 54 / 10000 = 0.0054$
Calculating the Total Sum
Now, we add the values of each term:
Sum = $5.4 + 0.54 + 0.054 + 0.0054$
To perform the addition, align the decimal points:
| $5.4000$ |
| $0.5400$ |
| $0.0540$ |
| $0.0054$ |
| ------- |
| $5.9994$ |
The sum is $5.9994$. This matches Option B.
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