Question

b find each quotient.
12\sqrt{30} 14\sqrt{45} 18\sqrt{63}
13\sqrt{55} 15\sqrt{75} 21\sqrt{85}
c circle the tables that show an additive
pattern (y = x + a).

Explanation:

Step1: Analyze the first table

For the first table:
When \(x = 2,y=3\), \(y=x + 1\); when \(x = 4,y = 5\), \(y=x+1\); when \(x=6,y = 7\), \(y=x + 1\); when \(x=8,y=9\), \(y=x + 1\). It follows \(y=x + a\) (\(a = 1\)).

Step2: Analyze the second table

For the second table:
When \(x = 1,y=7\), \(y = 7x\); when \(x=2,y = 14\), \(y=7x\); when \(x = 3,y=21\), \(y=7x\); when \(x=4,y=28\), \(y=7x\). It follows \(y=kx\) (\(k = 7\)), not \(y=x + a\).

Step3: Analyze the third table

For the third table:
When \(x = 5,y=8\), \(y=x + 3\); when \(x=10,y = 13\), \(y=x+3\); when \(x=15,y=18\), \(y=x + 3\); when \(x=20,y=23\), \(y=x+3\). It follows \(y=x + a\) (\(a = 3\)).

Step4: Analyze the fourth table

For the fourth table:
When \(x = 0,y=5\), \(y=x + 5\); when \(x=1,y = 6\), \(y=x+5\); when \(x=2,y=7\), \(y=x + 5\). It follows \(y=x + a\) (\(a = 5\)).

Step5: Analyze the fifth table

For the fifth table:
When \(x = 4,y=8\), \(y = 2x\); when \(x=8,y = 16\), \(y=2x\); when \(x=12,y=24\), \(y=2x\). It follows \(y=kx\) (\(k = 2\)), not \(y=x + a\).

Step6: Analyze the sixth table

For the sixth table:
When \(x = 3,y=7\), \(y=x + 4\); when \(x=6,y = 10\), \(y=x + 4\). But if \(x\) increases by \(3\), \(y\) increases by \(3\) (from \(7\) to \(10\)), and if we assume \(y=x + 4\), when \(x = 9\) (assuming the next \(x\) value following the \(x\) - increment of \(3\)), \(y=x+4=13\). But the relationship is not consistent for all possible \(x\) - values in a general additive \(y=x + a\) sense (the \(x\) - increment is not \(1\) and we can't confirm a single \(a\) for all \(x\) - \(y\) pairs in a strict \(y=x + a\) form as per the problem's requirement of the pattern \(y=x + a\) (where \(a\) is a constant and \(x\) changes by \(1\) in a simple sense, here \(x\) changes by \(3\) which is a deviation from the basic \(y=x + a\) where \(x\) is incremented by \(1\) each time)).

Answer:

The first, third and fourth tables should be circled.