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 $316.22776601\dots$

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.