Simplify square root of 10^5

Solution:

Step 1:

Calculate $10^5$ to find the number under the square root: $\sqrt{10^5}$.

Step 2:

Express $10^5$ as a product of a perfect square and another term: $10^5 = (10^2)^2 \cdot 10$.

Step 2.1:

Isolate the perfect square part from the expression: $\sqrt{(10^2)^2 \cdot 10}$.

Step 2.2:

Recognize that $(10^2)^2$ is a perfect square: $\sqrt{(10^2)^2 \cdot 10}$.

Step 3:

Extract the square root of the perfect square outside the radical: $10^2 \sqrt{10}$.

Step 4:

Present the simplified result in its various forms:

• Exact Form: $10^2 \sqrt{10}$
• Decimal Form: Approximately $316.22776601 \ldots$

Knowledge Notes:

To simplify the square root of $10^5$, we use the following knowledge points:

1. Exponentiation: Raising a number to a power, such as $10^5$, means multiplying the number by itself the specified number of times.

2. Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, $\sqrt{100}$ is $10$ because $10 \times 10 = 100$.

3. Perfect Squares: These are numbers that are squares of integers. For example, $100$ is a perfect square because it is $10^2$.

4. Simplifying Square Roots: To simplify a square root, one can factor the number under the radical into a product of a perfect square and another term. The square root of the perfect square can be taken out of the radical, simplifying the expression.

5. Radical Properties: The square root of a product can be expressed as the product of the square roots, i.e., $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, provided that $a$ and $b$ are non-negative.

Using these principles, we can simplify $\sqrt{10^5}$ by expressing it as $\sqrt{(10^2)^2 \cdot 10}$, then taking the square root of the perfect square $(10^2)^2$ out of the radical, which is $10^2$, and leaving the square root of $10$ inside the radical. This results in $10^2 \sqrt{10}$, which can also be expressed in decimal form.