Question

is the number $5 + \sqrt{28}$ rational or irrational?
rational
irrational
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Explanation:

Step1: Simplify \(\sqrt{28}\)

We know that \(28 = 4\times7\), so \(\sqrt{28}=\sqrt{4\times7}\). By the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a = 4\) and \(b=7\) and \(a\geq0,b\geq0\)), we have \(\sqrt{4\times7}=\sqrt{4}\times\sqrt{7}=2\sqrt{7}\).

Step2: Analyze the nature of \(\sqrt{7}\)

The number \(\sqrt{7}\) is an irrational number because 7 is a prime number and the square root of a non - perfect square positive integer is irrational. A rational number multiplied by an irrational number is irrational, so \(2\sqrt{7}\) is irrational (since 2 is rational and \(\sqrt{7}\) is irrational).

Step3: Analyze the nature of \(5 + 2\sqrt{7}\) (which is \(5+\sqrt{28}\))

A rational number (5) added to an irrational number (\(2\sqrt{7}\)) is irrational. So \(5+\sqrt{28}\) is irrational.

Answer:

irrational