Question

choose two correct answers that are like radicals after simplifying. √32x, √50x², √72x², √18n

Explanation:

Step1: Simplify \(\sqrt{32x}\)

Factor 32 into \(16\times2\), so \(\sqrt{32x}=\sqrt{16\times2x}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we get \(\sqrt{16}\times\sqrt{2x} = 4\sqrt{2x}\).

Step2: Simplify \(\sqrt{50x^{2}}\)

Factor 50 into \(25\times2\) and \(x^{2}\) is a perfect square, so \(\sqrt{50x^{2}}=\sqrt{25\times2\times x^{2}}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) and \(\sqrt{x^{2}} = |x|\) (for real \(x\), if we assume \(x\geq0\), then \(\sqrt{x^{2}}=x\)), we get \(\sqrt{25}\times\sqrt{2}\times\sqrt{x^{2}}=5x\sqrt{2}\).

Step3: Simplify \(\sqrt{72x^{2}}\)

Factor 72 into \(36\times2\) and \(x^{2}\) is a perfect square, so \(\sqrt{72x^{2}}=\sqrt{36\times2\times x^{2}}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) and \(\sqrt{x^{2}} = |x|\) (assuming \(x\geq0\), \(\sqrt{x^{2}}=x\)), we get \(\sqrt{36}\times\sqrt{2}\times\sqrt{x^{2}} = 6x\sqrt{2}\).

Step4: Simplify \(\sqrt{18n}\)

Factor 18 into \(9\times2\), so \(\sqrt{18n}=\sqrt{9\times2n}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\), we get \(\sqrt{9}\times\sqrt{2n}=3\sqrt{2n}\).

Now, looking at the simplified forms: \(\sqrt{50x^{2}} = 5x\sqrt{2}\) and \(\sqrt{72x^{2}}=6x\sqrt{2}\) have the same radical part \(\sqrt{2}\) (after considering the variable part, but the radical part's non - variable part is the same). Wait, no, wait, let's re - check. Wait, \(\sqrt{50x^{2}} = 5|x|\sqrt{2}\) and \(\sqrt{72x^{2}}=6|x|\sqrt{2}\). If we assume \(x\geq0\), then they are \(5x\sqrt{2}\) and \(6x\sqrt{2}\), which are like radicals (same radical part \(\sqrt{2}\) and the same type of variable factor). Also, \(\sqrt{32x}=4\sqrt{2x}\) and \(\sqrt{18n}=3\sqrt{2n}\) are like radicals (same radical part \(\sqrt{2}\) with variable \(x\) and \(n\) respectively, but if we consider the form, they have the same radical structure). But let's check the problem again. Wait, maybe I made a mistake. Wait, \(\sqrt{50x^{2}}=5x\sqrt{2}\), \(\sqrt{72x^{2}} = 6x\sqrt{2}\), these two have the same radical part \(\sqrt{2}\) and the same variable factor \(x\) (assuming \(x\) is a non - negative real number). And \(\sqrt{32x}=4\sqrt{2x}\), \(\sqrt{18n}=3\sqrt{2n}\) have the same radical part \(\sqrt{2}\) with variables \(x\) and \(n\) respectively. But the problem says "choose two correct answers that are like radicals after simplifying". Let's re - simplify:

Wait, \(\sqrt{50x^{2}}=\sqrt{25\times2\times x^{2}} = 5|x|\sqrt{2}\)

\(\sqrt{72x^{2}}=\sqrt{36\times2\times x^{2}}=6|x|\sqrt{2}\)

These two have the same radical part \(\sqrt{2}\) and the same variable factor (the \(x\) part, since \(|x|\) is a factor). So they are like radicals.

Also, \(\sqrt{32x}=\sqrt{16\times2x}=4\sqrt{2x}\)

\(\sqrt{18n}=\sqrt{9\times2n}=3\sqrt{2n}\)

These two have the same radical part \(\sqrt{2}\) with variables \(x\) and \(n\) respectively. But maybe the intended answer is \(\sqrt{50x^{2}}\) and \(\sqrt{72x^{2}}\) or \(\sqrt{32x}\) and \(\sqrt{18n}\). Let's check the simplification again.

For \(\sqrt{50x^{2}}\): \(\sqrt{50x^{2}}=\sqrt{25\times2\times x^{2}} = 5x\sqrt{2}\) (assuming \(x\geq0\))

For \(\sqrt{72x^{2}}\): \(\sqrt{72x^{2}}=\sqrt{36\times2\times x^{2}}=6x\sqrt{2}\) (assuming \(x\geq0\))

These two have the same radical part \(\sqrt{2}\) and the same variable \(x\) multiplied, so they are like radicals.

For \(\sqrt{32x}\): \(4\sqrt{2x}\)

For \(\sqrt{18n}\): \(3\sqrt{2n}\)

These two have the same radical part \(\sqrt{2}\) with variables \(x\) and \(n\), so they are like radicals.

But the problem has four options? Wait, the image shows four? Wait, no, the user's image shows \(\sqrt{32x}\), \(\sqrt{50x^{2}}\), \(\sqrt{72x^{2}}\), and \(\sqrt{18n}\). So we need to find two that are like radicals after simplifying.

Let's re - express:

- \(\sqrt{32x}=4\sqrt{2x}\)
- \(\sqrt{50x^{2}} = 5x\sqrt{2}\)
- \(\sqrt{72x^{2}}=6x\sqrt{2}\)
- \(\sqrt{18n}=3\sqrt{2n}\)

So \(\sqrt{50x^{2}}\) and \(\sqrt{72x^{2}}\) are like radicals (same radical part \(\sqrt{2}\), same variable factor \(x\)), and \(\sqrt{32x}\) and \(\sqrt{18n}\) are like radicals (same radical part \(\sqrt{2}\), variables \(x\) and \(n\) which are both single - variable under the radical). But maybe the intended answer is \(\sqrt{50x^{2}}\) and \(\sqrt{72x^{2}}\) or \(\sqrt{32x}\) and \(\sqrt{18n}\). Let's check the problem statement again. It says "choose two correct answers that are like radicals after simplifying". So let's confirm:

Like radicals are radicals that have the same index (here, index 2, square roots) and the same radicand (the expression inside the radical) when simplified.

For \(\sqrt{50x^{2}}\) and \(\sqrt{72x^{2}}\):

Simplified, they are \(5x\sqrt{2}\) and \(6x\sqrt{2}\). The radicand part (the part under the square root after factoring out perfect squares) is \(\sqrt{2}\), and the non - radical part has the same variable \(x\). So they are like radicals.

For \(\sqrt{32x}\) and \(\sqrt{18n}\):

Simplified, they are \(4\sqrt{2x}\) and \(3\sqrt{2n}\). The radicand part is \(\sqrt{2x}\) and \(\sqrt{2n}\), which are similar in form (same radical structure with a single variable). So they are like radicals.

But maybe the problem expects \(\sqrt{50x^{2}}\) and \(\sqrt{72x^{2}}\) as the answer. Let's proceed with that.

Answer:

The two correct answers are \(\sqrt{50x^{2}}\) and \(\sqrt{72x^{2}}\) (or \(\sqrt{32x}\) and \(\sqrt{18n}\), but based on the simplification where the radical part is \(\sqrt{2}\) with the same variable \(x\), \(\sqrt{50x^{2}}\) and \(\sqrt{72x^{2}}\) are like radicals). So the two options are \(\sqrt{50x^{2}}\) and \(\sqrt{72x^{2}}\).