Finding the Square Root of $23 - 4\sqrt{15}$

We are asked to find the square root of the expression $23 - 4\sqrt{15}$. Finding the square root means finding a value that, when multiplied by itself, equals the original expression $23 - 4\sqrt{15}$.

Checking the Options by Squaring

One way to solve this type of problem, especially in a multiple-choice format, is to test each option by squaring it. We are looking for an option which, when squared, results in $23 - 4\sqrt{15}$.

Let's consider the options provided. We will specifically look at the option $\sqrt{3} - 2\sqrt{5}$, as it is indicated as relevant to the solution.

Calculating $(\sqrt{3} - 2\sqrt{5})^2$

We will square the expression $\sqrt{3} - 2\sqrt{5}$ using the algebraic identity for squaring a binomial: $(a-b)^2 = a^2 - 2ab + b^2$.

In this case, we can consider $a = \sqrt{3}$ and $b = 2\sqrt{5}$.

Let's calculate each term:

• The first term squared: $a^2 = (\sqrt{3})^2 = 3$
• Twice the product of the two terms: $2ab = 2 \times (\sqrt{3}) \times (2\sqrt{5}) = 2 \times 2 \times \sqrt{3 \times 5} = 4\sqrt{15}$
• The second term squared: $b^2 = (2\sqrt{5})^2 = (2)^2 \times (\sqrt{5})^2 = 4 \times 5 = 20$

Now, substitute these results back into the formula $(a-b)^2 = a^2 - 2ab + b^2$:

$(\sqrt{3} - 2\sqrt{5})^2 = (\sqrt{3})^2 - 2(\sqrt{3})(2\sqrt{5}) + (2\sqrt{5})^2$

$(\sqrt{3} - 2\sqrt{5})^2 = 3 - 4\sqrt{15} + 20$

Combine the numerical terms:

$(\sqrt{3} - 2\sqrt{5})^2 = (3 + 20) - 4\sqrt{15}$

$(\sqrt{3} - 2\sqrt{5})^2 = 23 - 4\sqrt{15}$

Verification of the Square Root

We have shown that squaring the expression $\sqrt{3} - 2\sqrt{5}$ results in $23 - 4\sqrt{15}$. This confirms that $\sqrt{3} - 2\sqrt{5}$ is indeed a square root of $23 - 4\sqrt{15}$.

General Method for Simplifying Nested Radicals

Expressions like $\sqrt{a \pm \sqrt{b}}$ or $\sqrt{a \pm c\sqrt{d}}$ can often be simplified if the expression inside the square root can be written as a perfect square, typically in the form $(\sqrt{x} \pm \sqrt{y})^2 = (x+y) \pm 2\sqrt{xy}$.

Our expression is $\sqrt{23 - 4\sqrt{15}}$. We can rewrite $4\sqrt{15}$ to fit the $2\sqrt{xy}$ part of the formula. $4\sqrt{15} = 2 \times (2\sqrt{15})$. To get the '2' inside the square root, we square it: $2\sqrt{15} = \sqrt{2^2 \times 15} = \sqrt{4 \times 15} = \sqrt{60}$.

So, $\sqrt{23 - 4\sqrt{15}} = \sqrt{23 - 2\sqrt{60}}$.

Now we need to find two numbers, say $x$ and $y$, such that their sum $x+y = 23$ and their product $xy = 60$. We look for pairs of factors of 60:

The pair (3, 20) satisfies both conditions: $3+20=23$ and $3 \times 20 = 60$.

So, $23 - 2\sqrt{60}$ can be written as $20 + 3 - 2\sqrt{20 \times 3}$. This matches the form $(\sqrt{x} - \sqrt{y})^2$ where $x=20$ and $y=3$.

Thus, $\sqrt{23 - 2\sqrt{60}} = \sqrt{(\sqrt{20} - \sqrt{3})^2}$ or $\sqrt{(\sqrt{3} - \sqrt{20})^2}$.

This simplifies to $|\sqrt{20} - \sqrt{3}|$ or $|\sqrt{3} - \sqrt{20}|$.

Since $\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$, the result is $|2\sqrt{5} - \sqrt{3}|$ or $|\sqrt{3} - 2\sqrt{5}|$.

The absolute value $|z|$ is $z$ if $z \ge 0$ and $-z$ if $z < 0$. Since $2\sqrt{5} = \sqrt{20}$ and $\sqrt{3}$, and $20 > 3$, $2\sqrt{5} > \sqrt{3}$. Therefore, $2\sqrt{5} - \sqrt{3}$ is positive, and its absolute value is $2\sqrt{5} - \sqrt{3}$. This is the principal (non-negative) square root.

The expression $\sqrt{3} - 2\sqrt{5}$ is negative because $\sqrt{3} < 2\sqrt{5}$. Its absolute value is $-( \sqrt{3} - 2\sqrt{5} ) = 2\sqrt{5} - \sqrt{3}$.

Both $\sqrt{3} - 2\sqrt{5}$ and $2\sqrt{5} - \sqrt{3}$ are square roots of $23 - 4\sqrt{15}$ because squaring either expression gives $23 - 4\sqrt{15}$. The options provided include $\sqrt{3} - 2\sqrt{5}$, which we confirmed by squaring.

Revision Table: Simplifying Radical Expressions

Additional Information: Understanding Square Roots

A square root of a number $N$ is a number that, when squared (multiplied by itself), gives $N$. Every positive number has two square roots, one positive and one negative.

For example, the square roots of 9 are 3 and -3, because $3^2 = 9$ and $(-3)^2 = 9$.

The symbol $\sqrt{}$ is typically used to denote the principal square root, which is defined as the non-negative square root. So, $\sqrt{9} = 3$.

In this problem, $23 - 4\sqrt{15}$ is a positive value. Its principal square root is the positive value $2\sqrt{5} - \sqrt{3}$. However, the question asks for "the square root" and provides $\sqrt{3} - 2\sqrt{5}$ as one of the options, which is a negative number. Since $(\sqrt{3} - 2\sqrt{5})^2 = 23 - 4\sqrt{15}$, $\sqrt{3} - 2\sqrt{5}$ is also a square root of $23 - 4\sqrt{15}$. When asked to select from options, the option that squares back to the original value is considered correct.