Understanding the Concept of 1000.5 and Its Equalities to 10

The expression 1000.5 represents the square root of 100. In mathematical terms, raising a number to the power of 0.5 is equivalent to taking the square root of that number.

Mathematical Definition and Derivation

The square root of a number x is defined as a value y such that y2 x.

For x 100, we need to find y such that y2 100. The value that satisfies this equation is y 10 because 102 100.

Thus, we can conclude that: 1000.5 radic;100 10.

Essentially, 1000.5 is equal to 1001/2, which is the same as radic;100, and radic;100 is equal to 10. Taking the power of 1/2 means taking the square root, and since 1010 100, the square root of 100 is 10.

Mathematical Explanation

To further clarify, let's delve into the math behind this. The expression 1002 means 100 times 100, while 1003 means 100 times 100 times 100. In general, an a times; a times; ... times; a, where we have n a's in this multiplication.

When dealing with exponents, the rule am times; an amn helps us to understand how to extend the concept to non-integer exponents. For a1/2, we need to find a value that, when multiplied by itself, equals a. This value is defined as the square root of a, i.e., a1/2 radic;a.

Applying this to 100, we know that 1001/2 is such that (1001/2)2 100. Therefore, 1001/2 10.

From the rule am times; an amn, if m 1/2, then a1/2 times; a1/2 a1 a. Hence, if we let x a1/2, we have x2 a. This implies that x radic;a, or a1/2 radic;a.

Therefore, 1001/2 10, because 1001/2 radic;100, and radic;100 10.

Conclusion

In conclusion, the expression 1000.5 equals 10 because the square root of 100 is defined as 10. This is based on the fundamental mathematical principles of exponents and square roots, which we have explored in depth to ensure clarity and understanding.