Simplify | square root of 11+i square root of 5|

Solution:

Step:1 Switch the positions of $i$ and $\sqrt{5}$ to match the standard complex number format.$\left|\right. \sqrt{11} + i\sqrt{5} \left|\right.$

Step:2 Apply the modulus formula for a complex number $\left|\right. a + bi \left|\right. = \sqrt{a^{2} + b^{2}}$ to calculate the magnitude.$\sqrt{\left(\sqrt{11}\right)^{2} + \left(\sqrt{5}\right)^{2}}$

Step:3 Simplify $\left(\sqrt{11}\right)^{2}$ to $11$.

Step:3.1 Express $\sqrt{11}$ using exponent notation as $\left(11\right)^{\frac{1}{2}}$.$\sqrt{\left(\left(11\right)^{\frac{1}{2}}\right)^{2} + \left(\sqrt{5}\right)^{2}}$

Step:3.2 Utilize the exponent multiplication rule, which states $\left(a^{m}\right)^{n} = a^{m \cdot n}$.$\sqrt{11^{\frac{1}{2} \cdot 2} + \left(\sqrt{5}\right)^{2}}$

Step:3.3 Combine the exponents $\frac{1}{2}$ and $2$.$\sqrt{11^{\frac{2}{2}} + \left(\sqrt{5}\right)^{2}}$

Step:3.4 Eliminate the common factor of $2$.

Step:3.4.1 Remove the common factor from the exponent.$\sqrt{11^{\frac{\cancel{2}}{\cancel{2}}} + \left(\sqrt{5}\right)^{2}}$

Step:3.4.2 Rewrite the simplified expression.$\sqrt{11^{1} + \left(\sqrt{5}\right)^{2}}$

Step:3.5 Evaluate the exponent to its simplest form.$\sqrt{11 + \left(\sqrt{5}\right)^{2}}$

Step:4 Simplify $\left(\sqrt{5}\right)^{2}$ to $5$.

Step:4.1 Express $\sqrt{5}$ using exponent notation as $5^{\frac{1}{2}}$.$\sqrt{11 + \left(5^{\frac{1}{2}}\right)^{2}}$

Step:4.2 Apply the exponent multiplication rule.$\sqrt{11 + 5^{\frac{1}{2} \cdot 2}}$

Step:4.3 Combine the exponents $\frac{1}{2}$ and $2$.$\sqrt{11 + 5^{\frac{2}{2}}}$

Step:4.4 Eliminate the common factor of $2$.

Step:4.4.1 Remove the common factor from the exponent.$\sqrt{11 + 5^{\frac{\cancel{2}}{\cancel{2}}}}$

Step:4.4.2 Rewrite the simplified expression.$\sqrt{11 + 5^{1}}$

Step:4.5 Evaluate the exponent to its simplest form.$\sqrt{11 + 5}$

Step:5 Final simplification of the expression.

Step:5.1 Combine $11$ and $5$.$\sqrt{16}$

Step:5.2 Express $16$ as the square of $4$, which is $4^{2}$.$\sqrt{4^{2}}$

Step:5.3 Extract the square root of a perfect square, assuming positive real numbers.$4$

Knowledge Notes:

1. Complex Numbers: A complex number is of the form $a + bi$ where $a$ is the real part, $b$ is the imaginary part, and $i$ is the imaginary unit with the property $i^2 = -1$.

2. Modulus of a Complex Number: The modulus of a complex number $a + bi$ is given by $\sqrt{a^2 + b^2}$, representing the distance of the complex number from the origin in the complex plane.

3. Exponent Rules: The power of a power rule states that $(a^m)^n = a^{m \cdot n}$. Also, $a^1 = a$ and $a^0 = 1$ for any non-zero $a$.

4. Square Roots and Exponents: The square root of a number $a$ can be written as $a^{\frac{1}{2}}$. Squaring the square root of a number returns the original number, i.e., $(\sqrt{a})^2 = a$.

5. Simplifying Expressions: The process of simplifying expressions involves applying arithmetic operations and algebraic rules to rewrite expressions in a simpler or more convenient form.

6. Perfect Squares: A perfect square is a number that is the square of an integer. The square root of a perfect square is an integer. For example, $16$ is a perfect square because it is $4^2$, and its square root is $4$.