Option A: [tex]\sqrt{75}[/tex]
Option C: [tex]\sqrt{15} \cdot \sqrt{5}[/tex]
Option F: [tex]\sqrt{25} \cdot \sqrt{3}[/tex]
Solution:
Given expression is [tex]5 \sqrt{3}[/tex].
Option A: [tex]\sqrt{75}[/tex]
[tex]\sqrt{75}=\sqrt{25\times3}[/tex]
[tex]=\sqrt{5^2\times3}[/tex]
[tex]=5\sqrt{3}[/tex]
Hence [tex]\sqrt{75}[/tex] is equivalent expression of [tex]5 \sqrt{3}[/tex].
Option B: [tex]\sqrt{45}[/tex]
[tex]\sqrt{45}=\sqrt{9\times5}[/tex]
[tex]=\sqrt{3^2\times5}[/tex]
[tex]=3\sqrt{5}[/tex]
Hence [tex]\sqrt{45}[/tex] is not equivalent expression of [tex]5 \sqrt{3}[/tex].
Option C: [tex]\sqrt{15} \cdot \sqrt{5}[/tex]
[tex]\sqrt{15} \cdot \sqrt{5}=\sqrt{15\times5}[/tex]
[tex]=\sqrt{75}[/tex]
[tex]=5\sqrt{3}[/tex] (proved in option A)
Hence [tex]\sqrt{15} \cdot \sqrt{5}[/tex] is equivalent expression of [tex]5 \sqrt{3}[/tex].
Option D: [tex]\sqrt{3} \cdot \sqrt{5}[/tex]
[tex]\sqrt{3} \cdot \sqrt{5}=\sqrt{3\times5}[/tex]
[tex]=\sqrt{15}[/tex]
Hence [tex]\sqrt{3} \cdot \sqrt{5}[/tex] is not equivalent expression of [tex]5 \sqrt{3}[/tex].
Option E: 75
75 is a whole number.
Hence 75 is not equivalent expression of [tex]5 \sqrt{3}[/tex].
Option F: [tex]\sqrt{25} \cdot \sqrt{3}[/tex]
[tex]\sqrt{25} \cdot \sqrt{3}=\sqrt{25\times3}[/tex]
[tex]=\sqrt{75}[/tex]
[tex]=5\sqrt{3}[/tex] (proved in option A)
Hence [tex]\sqrt{25} \cdot \sqrt{3}[/tex] is equivalent expression of [tex]5 \sqrt{3}[/tex].
Therefore, [tex]\sqrt{75},\ \ \sqrt{15} \cdot \sqrt{5}, \ \ \sqrt{25} \cdot \sqrt{3}[/tex] are all equivalent expressions of [tex]5 \sqrt{3}[/tex].
