Question

which value is an irrational number?

a ( 4 + sqrt{7} )

b ( sqrt{2}sqrt{8} )

c ( \frac{sqrt{3}sqrt{12}}{5} )

d ( sqrt{3} - sqrt{3} )

Explanation:

Step1: Analyze Option a

\(\sqrt{7}\) is an irrational number (since 7 is not a perfect square). The sum of a rational number (4) and an irrational number (\(\sqrt{7}\)) is irrational. So \(4+\sqrt{7}\) is irrational.

Step2: Analyze Option b

Simplify \(\sqrt{2}\sqrt{8}\). Using the property \(\sqrt{a}\sqrt{b}=\sqrt{ab}\), we get \(\sqrt{2\times8}=\sqrt{16} = 4\), which is rational.

Step3: Analyze Option c

Simplify \(\frac{\sqrt{3}\sqrt{12}}{5}\). First, \(\sqrt{3}\sqrt{12}=\sqrt{3\times12}=\sqrt{36}=6\). Then \(\frac{6}{5}\) is rational.

Step4: Analyze Option d

\(\sqrt{3}-\sqrt{3}=0\), which is a rational number (an integer).

Answer:

a. \( 4 + \sqrt{7} \)