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
