Question

the square root function
quiz complete
1 2 3 4 5
what is the simplified form of \\(\sqrt{144x^{36}}\\)?
\\(\circ\\) \\(12x^6\\)
\\(\circ\\) \\(12x^{18}\\)
\\(\circ\\) \\(72x^6\\)
\\(\circ\\) \\(72x^{18}\\)

Explanation:

Step1: Simplify the square root of 144

The square root of 144 is 12, since \(12\times12 = 144\). So, \(\sqrt{144}=12\).

Step2: Simplify the square root of \(x^{36}\)

Using the property of exponents \(\sqrt{a^m}=a^{\frac{m}{2}}\), for \(a = x\) and \(m = 36\), we have \(\sqrt{x^{36}}=x^{\frac{36}{2}}=x^{18}\)? Wait, no, wait. Wait, the original expression is \(\sqrt{144x^{36}}\), which can be written as \(\sqrt{144}\times\sqrt{x^{36}}\). Wait, no, wait, maybe I made a mistake. Wait, \(x^{36}\) under the square root: \(\sqrt{x^{36}}=x^{18}\)? But wait, the options have \(12x^{18}\) and \(12x^6\). Wait, no, wait, maybe the original problem is \(\sqrt{144x^{12}}\)? Wait, no, the user wrote \(x^{36}\). Wait, no, let's check again. Wait, \(\sqrt{144x^{36}}=\sqrt{144}\times\sqrt{x^{36}} = 12\times x^{18}\), because \(\sqrt{x^{36}}=x^{36/2}=x^{18}\). But wait, the options include \(12x^{18}\) as option B. Wait, but maybe I misread the exponent. Wait, maybe the exponent is \(x^{12}\)? No, the user's image shows \(x^{36}\). Wait, but let's check the options. The options are:

• 12x⁶

• 12x¹⁸

• 72x⁶

• 72x¹⁸

Wait, so according to the calculation, \(\sqrt{144x^{36}} = 12x^{18}\), which is option B. Wait, but maybe the original problem was \(x^{12}\)? No, the user's image shows \(x^{36}\). Wait, maybe I made a mistake in the exponent. Wait, no, \(\sqrt{x^{36}} = x^{18}\), so 12x¹⁸ is the correct simplification.

Wait, but let's re - check. The square root of a product is the product of the square roots: \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)). So \(\sqrt{144x^{36}}=\sqrt{144}\times\sqrt{x^{36}}\). \(\sqrt{144} = 12\), and \(\sqrt{x^{36}}=x^{36/2}=x^{18}\). So the simplified form is \(12x^{18}\).

Answer:

B. \(12x^{18}\)