Evaluate square root of 7^2+1^2

Solution:

Step 1:

Square the number $7$. $\sqrt{49 + 1^2}$

Step 2:

Any number raised to the power of one remains the same. $\sqrt{49 + 1}$

Step 3:

Combine $49$ and $1$. $\sqrt{50}$

Step 4:

Express $50$ as the product of $5^2$ and $2$.

Step 4.1:

Extract the square of $5$ from the product. $\sqrt{25 \cdot 2}$

Step 4.2:

Represent $25$ as $5^2$. $\sqrt{5^2 \cdot 2}$

Step 5:

Remove the square root from the squared term. $5\sqrt{2}$

Step 6:

Present the solution in its various forms.

Exact Form: $5\sqrt{2}$ Decimal Form: Approximately $7.07106781$

Knowledge Notes:

To solve a problem involving the square root of the sum of squares, we follow these steps:

1. Squaring Numbers: When a number is squared, it is multiplied by itself. For example, $7^2 = 7 \times 7 = 49$.

2. Exponent Laws: Any number raised to the power of $1$ remains unchanged, as $n^1 = n$ for any number $n$.

3. Simplifying Square Roots: When simplifying the square root of a product, if one of the factors is a perfect square, it can be taken out of the square root as its base. For example, $\sqrt{a^2 \cdot b} = a\sqrt{b}$ if $a$ is an integer.

4. Combining Like Terms: Arithmetic operations such as addition must be performed before taking the square root.

5. Square Root of a Product: The square root of a product $\sqrt{a \cdot b}$ can be expressed as $\sqrt{a} \cdot \sqrt{b}$ if both $a$ and $b$ are non-negative.

6. Exact and Decimal Forms: The exact form of a square root is the simplified radical expression, while the decimal form is the approximate numerical value, which can be found using a calculator.

In this problem, we use these principles to simplify the expression $\sqrt{7^2 + 1^2}$, ultimately finding that it equals $5\sqrt{2}$ in exact form and approximately $7.07106781$ in decimal form.