Simplify: \(\sqrt {176 + \sqrt {2401} } \)
15
Simplifying Nested Square Roots: Step-by-Step Guide
The problem asks us to simplify the mathematical expression involving nested square roots: \(\sqrt {176 + \sqrt {2401} }\). To solve this, we need to work from the inside out, first evaluating the innermost square root and then proceeding with the outer expression.
Step 1: Evaluate the Innermost Square Root
The innermost part of the expression is \(\sqrt{2401}\). We need to find the number that, when multiplied by itself, gives 2401.
- We can estimate the square root. We know \(40^2 = 1600\) and \(50^2 = 2500\). So, the square root of 2401 is between 40 and 50.
- The number 2401 ends with the digit 1. This means its square root must end with either 1 (since \(1^2=1\)) or 9 (since \(9^2=81\)).
- Considering our estimate, the possible values are 41 or 49.
- Let's check 49: \(49 \times 49\).
Calculation:
49 x 49 ---- 441 (9 * 49) 1960 (40 * 49) ---- 2401
So, \(\sqrt{2401} = 49\).
Step 2: Substitute the Value and Simplify the Expression
Now we substitute the value of \(\sqrt{2401}\) back into the original expression:
The expression becomes \(\sqrt{176 + 49}\).
Next, we need to calculate the sum inside the square root:
\(176 + 49 = 225\).
So the expression simplifies to \(\sqrt{225}\).
Step 3: Evaluate the Outer Square Root
Finally, we need to find the square root of 225. We need the number that, when multiplied by itself, gives 225.
- We know \(10^2 = 100\) and \(20^2 = 400\). So, the square root of 225 is between 10 and 20.
- The number 225 ends with the digit 5. This means its square root must end with 5 (since \(5^2=25\)).
- The only number between 10 and 20 that ends in 5 is 15.
- Let's check 15: \(15 \times 15 = 225\).
So, \(\sqrt{225} = 15\).
Therefore, the simplified value of the expression \(\sqrt {176 + \sqrt {2401} }\) is 15.
Summary of Simplification Steps
Here's a quick recap of the simplification process:
- Identify the innermost square root: \(\sqrt{2401}\).
- Calculate its value: \(\sqrt{2401} = 49\).
- Substitute the value back: \(\sqrt{176 + 49}\).
- Calculate the sum inside the square root: \(176 + 49 = 225\).
- Evaluate the remaining square root: \(\sqrt{225} = 15\).
| Step | Operation | Calculation | Result |
|---|---|---|---|
| 1 | Innermost square root | \(\sqrt{2401}\) | 49 |
| 2 | Substitution and Addition | \(176 + 49\) | 225 |
| 3 | Outer square root | \(\sqrt{225}\) | 15 |
Revision Table: Key Concepts for Simplifying Square Roots
| Concept | Description | Example |
|---|---|---|
| Square Root | A number that produces a given number when multiplied by itself. Represented by the symbol \(\sqrt{}\). | \(\sqrt{9} = 3\) because \(3 \times 3 = 9\) |
| Perfect Square | An integer that is the square of an integer. | 9, 16, 25, 49, 225, 2401 are perfect squares. |
| Simplifying Nested Radicals | Evaluating the expression by working from the innermost radical outwards. | \(\sqrt{a + \sqrt{b}}\) requires finding \(\sqrt{b}\) first, then calculating \(a + \sqrt{b}\), then taking the square root of the result. |
Additional Information: Finding Square Roots
Finding the square root of a large number like 2401 can be done using various methods:
- Estimation: Bounding the number between squares of tens (\(40^2=1600, 50^2=2500\)).
- Unit Digit Analysis: Looking at the last digit of the number (ends in 1, so root ends in 1 or 9).
- Prime Factorization: Breaking the number down into its prime factors. For 2401, \(2401 = 7 \times 343 = 7 \times 7 \times 49 = 7 \times 7 \times 7 \times 7 = 7^4\). Then \(\sqrt{2401} = \sqrt{7^4} = 7^{4/2} = 7^2 = 49\).
- Long Division Method: A systematic algorithm similar to long division for finding square roots.
For common perfect squares like 225, it's helpful to memorize them up to a certain point (e.g., up to \(20^2\) or \(25^2\)) as they appear frequently in problems.
Important Questions from Surds and Indices
The value of (0.3) [{(200 - 146)/(3 × 3 × 3)} - 3] is:
The expression \(\frac{{15\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\sqrt {10\;} + \sqrt {20} + \sqrt {40} - \sqrt 5 - \sqrt {80} }}\) is equal to:
Let \(x = \left( {\frac{{√ {1875} }}{{√ {3888} }} \div \frac{{√ {1200} }}{{\sqrt 768}}} \right) \times \frac{{√ {175} }}{{√ {1792} }}\) . Then √x is equal to:
If \(x = \sqrt {-\sqrt 3 + \sqrt {3 + 8\sqrt {7 + 4\sqrt 3}}}\) where x > 0, then the value of x is equal to:
What is the value of \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}−\sqrt{5}} \div \frac{\sqrt{14}+\sqrt{10}}{\sqrt{14}−\sqrt{10}}+\frac{\sqrt{10}}{\sqrt{5}}\) ?
