Simplify the Radical Expression square root of -20* square root of -5

Solution:

Step 1:

Express $-20$ as $-1 \times 20$. Thus, $\sqrt{-20} \cdot \sqrt{-5}$ becomes $\sqrt{-1 \times 20} \cdot \sqrt{-5}$.

Step 2:

Separate the square roots, $\sqrt{-1 \times 20}$ can be written as $\sqrt{-1} \cdot \sqrt{20}$. Now we have $\sqrt{-1} \cdot \sqrt{20} \cdot \sqrt{-5}$.

Step 3:

Recognize that $\sqrt{-1}$ is the imaginary unit $i$. The expression is now $i \cdot \sqrt{20} \cdot \sqrt{-5}$.

Step 4:

Similarly, express $-5$ as $-1 \times 5$ and rewrite the expression as $i \cdot \sqrt{20} \cdot \sqrt{-1 \times 5}$.

Step 5:

Again, separate the square roots, $\sqrt{-1 \times 5}$ becomes $\sqrt{-1} \cdot \sqrt{5}$. The expression is now $i \cdot \sqrt{20} \cdot (i \cdot \sqrt{5})$.

Step 6:

Replace $\sqrt{-1}$ with $i$ in the second part of the expression, leading to $i \cdot \sqrt{20} \cdot (i \cdot \sqrt{5})$.

Step 7:

Decompose 20 into its prime factors as $2^2 \times 5$.

Step 7.1:

Extract the square root of the perfect square, $4$, from $\sqrt{20}$, giving $i \cdot \sqrt{4 \times 5} \cdot (i \cdot \sqrt{5})$.

Step 7.2:

Rewrite $4$ as $2^2$, so the expression becomes $i \cdot \sqrt{2^2 \times 5} \cdot (i \cdot \sqrt{5})$.

Step 8:

Take the square root of the perfect square, $2^2$, out from under the radical, which becomes $i \cdot (2 \sqrt{5}) \cdot (i \cdot \sqrt{5})$.

Step 9:

Understand that the absolute value of $2$ is simply $2$, so the expression simplifies to $i \cdot (2 \sqrt{5}) \cdot (i \cdot \sqrt{5})$.

Step 10:

Rearrange to place $2$ before $i$, resulting in $2i \sqrt{5} \cdot (i \cdot \sqrt{5})$.

Step 11:

Multiply $2i \sqrt{5} \cdot (i \sqrt{5})$.

Step 11.1:

$i$ raised to the first power is $i$, so we have $2(i^1 \cdot i) \sqrt{5} \sqrt{5}$.

Step 11.2:

Again, $i$ raised to the first power is $i$, leading to $2(i^1 \cdot i^1) \sqrt{5} \sqrt{5}$.

Step 11.3:

Apply the exponent rule $a^{m} \cdot a^{n} = a^{m + n}$ to combine the $i$ exponents, resulting in $2i^{1 + 1} \sqrt{5} \sqrt{5}$.

Step 11.4:

Add the exponents of $i$, which gives $2i^2 \sqrt{5} \sqrt{5}$.

Step 11.5:

$\sqrt{5}$ raised to the first power remains $\sqrt{5}$, so we have $2i^2 (\sqrt{5})^1 \sqrt{5}$.

Step 11.6:

Again, $\sqrt{5}$ raised to the first power is $\sqrt{5}$, leading to $2i^2 ((\sqrt{5})^1 (\sqrt{5})^1)$.

Step 11.7:

Combine the exponents of $\sqrt{5}$ using the rule $a^{m} \cdot a^{n} = a^{m + n}$, resulting in $2i^2 (\sqrt{5})^{1 + 1}$.

Step 11.8:

Add the exponents of $\sqrt{5}$, which simplifies to $2i^2 (\sqrt{5})^2$.

Step 12:

$i^2$ is equal to $-1$, so the expression becomes $2 \cdot -1 (\sqrt{5})^2$.

Step 13:

Multiply $2$ by $-1$, resulting in $-2 (\sqrt{5})^2$.

Step 14:

$(\sqrt{5})^2$ is equal to $5$.

Step 14.1:

Rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$, so the expression is $-2 ((5^{\frac{1}{2}})^2)$.

Step 14.2:

Apply the power rule $(a^{m})^n = a^{m \cdot n}$, which gives $-2 \cdot 5^{\frac{1}{2} \cdot 2}$.

Step 14.3:

Multiply the exponents, resulting in $-2 \cdot 5^{\frac{2}{2}}$.

Step 14.4:

Simplify the exponent by canceling out the common factor of $2$.

Step 14.4.1:

Cancel the common factor to get $-2 \cdot 5^{\frac{\cancel{2}}{\cancel{2}}}$.

Step 14.4.2:

Rewrite the expression as $-2 \cdot 5^1$.

Step 14.5:

Evaluate the exponent to get $-2 \cdot 5$.

Step 15:

Finally, multiply $-2$ by $5$ to obtain $-10$.

Knowledge Notes:

• Imaginary Unit: The imaginary unit $i$ is defined as $\sqrt{-1}$. It is used to extend the real numbers to complex numbers.

• Radical Simplification: When simplifying radicals, one can separate the radical into the product of separate radicals if the original radicand is a product itself.

• Exponent Rules:

  • The power rule states that $a^{m} \cdot a^{n} = a^{m + n}$.

  • The power of a power rule states that $(a^{m})^{n} = a^{m \cdot n}$.

  • These rules are used to simplify expressions involving exponents.

• Absolute Value: The absolute value of a number is its distance from zero on the number line, regardless of direction. The absolute value of a positive number or zero is the number itself, and the absolute value of a negative number is its opposite.

• Complex Numbers: A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit. Complex numbers include the field of real numbers as a subset.

• Multiplying Complex Numbers: When multiplying complex numbers, apply the distributive property and use the fact that $i^2 = -1$ to simplify.

• Square Roots and Exponents: The square root of a number $x$ can be expressed as $x^{1/2}$. When raising a square root to the power of $2$, the result is the original number under the square root.