What is the square root of $16 x^{36}$ ?
The objective is to compute the square root of the expression \(16x^{36}\), where \(x\) is a real number. The square root function, denoted as \(\sqrt{\cdot}\), yields a non-negative value. To evaluate \(\sqrt{16x^{36}}\), we use the algebraic properties of square roots and exponents. Specifically, we recall the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) for \(a, b \geq 0\), and the power rule \((x^m)^n = x^{m \cdot n}\).
We can split the expression under the square root into two parts: the constant and the variable part. This gives:
\[\sqrt{16x^{36}} = \sqrt{16} \cdot \sqrt{x^{36}}.\]
First, we calculate the square root of the constant:
\[\sqrt{16} = 4.\]
Next, we simplify \(\sqrt{x^{36}}\). Using the power rule of exponents, we can express \(x^{36}\) as the square of another term:
\[x^{36} = (x^{18})^2.\]
Therefore, \(\sqrt{x^{36}} = \sqrt{(x^{18})^2} = |x^{18}|\). Since \(x^{18}\) is non-negative for all real numbers \(x\), we have \(|x^{18}| = x^{18}\). Combining these, the entire expression simplifies to:
\[4 \cdot x^{18} = 4x^{18}.\]
Comparing our calculated result \(4x^{18}\) with the given options:
The result matches option (c). Thus, the square root of \(16x^{36}\) is \(\boxed{4x^{18}}\).
In fields like engineering or physics, expressions involving high powers are common when dealing with area or volume formulas that change over time. For example, if the area of a square surface is given by \(16x^{36}\), finding the length of one side requires taking the square root. This calculation, \(\sqrt{16x^{36}} = 4x^{18}\), helps identify dimensions that are crucial for structural analysis or material estimation in practical design projects.