Answer :

Sure, let's find the simplest rationalizing factor for each of the given expressions.

### (a) [tex]\( \sqrt{28} \)[/tex]

To rationalize [tex]\( \sqrt{28} \)[/tex], we need to find a number such that when multiplied by [tex]\( \sqrt{28} \)[/tex], eliminates the square root.

The simplest factor to multiply by [tex]\( \sqrt{28} \)[/tex] is [tex]\( \sqrt{28} \)[/tex] itself:

[tex]\[ \sqrt{28} \times \sqrt{28} = 28 \][/tex]

So, the simplest rationalizing factor for [tex]\( \sqrt{28} \)[/tex] is 28.

### (b) [tex]\( \sqrt{75} \)[/tex]

Similarly, to rationalize [tex]\( \sqrt{75} \)[/tex], we need to multiply it by [tex]\( \sqrt{75} \)[/tex] to obtain a rational number:

[tex]\[ \sqrt{75} \times \sqrt{75} = 75 \][/tex]

Thus, the simplest rationalizing factor for [tex]\( \sqrt{75} \)[/tex] is 75.

### (c) [tex]\( 10 \sqrt{2} \)[/tex]

For [tex]\( 10 \sqrt{2} \)[/tex], we focus on the irrational part, which is [tex]\( \sqrt{2} \)[/tex]. To rationalize [tex]\( \sqrt{2} \)[/tex], we multiply it by [tex]\( \sqrt{2} \)[/tex]:

[tex]\[ 10 \sqrt{2} \times \sqrt{2} = 10 \times 2 = 20 \][/tex]

Thus, the simplest rationalizing factor for [tex]\( 10 \sqrt{2} \)[/tex] relative to its irrational part [tex]\( \sqrt{2} \)[/tex] is 2.

### (d) [tex]\( 3 \sqrt{98} \)[/tex]

For [tex]\( 3 \sqrt{98} \)[/tex], we need to rationalize [tex]\( \sqrt{98} \)[/tex]. The simplest way to achieve this is by multiplying by [tex]\( \sqrt{98} \)[/tex]:

[tex]\[ 3 \sqrt{98} \times \sqrt{98} = 3 \times 98 = 294 \][/tex]

Thus, the simplest rationalizing factor for [tex]\( 3 \sqrt{98} \)[/tex] relative to its irrational part [tex]\( \sqrt{98} \)[/tex] is 98.

So to summarize, the simplest rationalizing factors are:
- (a) 28
- (b) 75
- (c) 2
- (d) 98