If √0.00000676 = 0.0026, then what is the value of √6760000?

Introduction / Context:
This question relates two square roots whose numbers share the same digits but differ by their decimal placement. We are told the square root of a small decimal number and asked to find the square root of a large number constructed from the same digits. The problem tests understanding of how square roots behave with powers of 10.

Given Data / Assumptions:
- √0.00000676 = 0.0026 is given.
- We need √6760000.
- All numbers are positive real numbers.

Concept / Approach:
We use scientific notation to understand the relation between the two numbers. Multiplying a number by a power of 10 shifts the decimal point. The square root of 10ⁿ is 10^(n/2). If two numbers differ by a factor of 10ᵏ, then their square roots differ by a factor of 10^(k/2). We use the given square root to deduce the required one by scaling appropriately.

Step-by-Step Solution:
Step 1: Rewrite 0.00000676 in scientific form: 0.00000676 = 6.76 × 10⁻⁶.Step 2: Then √0.00000676 = √(6.76 × 10⁻⁶) = √6.76 × 10⁻³, which is given as 0.0026 = 2.6 × 10⁻³. So √6.76 = 2.6.Step 3: Now consider 6760000. Write 6760000 = 6.76 × 10⁶ (the same 6.76 with 10⁶ instead of 10⁻⁶).Step 4: Then √6760000 = √(6.76 × 10⁶) = √6.76 × 10³.Step 5: Since √6.76 = 2.6, we get √6760000 = 2.6 × 10³ = 2600.

Verification / Alternative check:
We can square 2600 to verify. 2600² = (26 × 100)² = 26² × 100² = 676 × 10⁴ = 6,760,000 which matches 6760000, confirming that 2600 is the correct square root.

Why Other Options Are Wrong:
2.6 and 26 are far too small; their squares are 6.76 and 676, respectively, which are nowhere near 6760000. The option 260 has square 67600, still smaller than the target by a factor of 100. Only 2600 gives the correct square when multiplied by itself.

Common Pitfalls:
A common error is to miscount decimal places when converting between forms like 0.00000676 and 6760000. Another mistake is to forget that when a number is multiplied by 10⁶, its square root is multiplied by 10³, not by 10⁶. Carefully handling the exponents of 10 avoids these issues.

Final Answer:
The value of √6760000 is 2600.