To simplify radicals, we need to find the largest perfect square that divides evenly into the number under the radical sign. Then we can pull its square root out in front.
Let’s go through each problem one by one:
1) $-3\sqrt{192}$
192 = 64 × 3 → √192 = √(64×3) = 8√3
So: -3 × 8√3 =
-24√3
2) $-5\sqrt{108}$
108 = 36 × 3 → √108 = 6√3
So: -5 × 6√3 =
-30√3
3) $-2\sqrt{50}$
50 = 25 × 2 → √50 = 5√2
So: -2 × 5√2 =
-10√2
4) $8\sqrt{128}$
128 = 64 × 2 → √128 = 8√2
So: 8 × 8√2 =
64√2
5) $-4\sqrt{72}$
72 = 36 × 2 → √72 = 6√2
So: -4 × 6√2 =
-24√2
6) $2\sqrt{98}$
98 = 49 × 2 → √98 = 7√2
So: 2 × 7√2 =
14√2
7) $4\sqrt{72}$
Same as #5 but positive: 4 × 6√2 =
24√2
8) $-4\sqrt{252}$
252 = 36 × 7 → √252 = 6√7
So: -4 × 6√7 =
-24√7
9) $-2\sqrt{28}$
28 = 4 × 7 → √28 = 2√7
So: -2 × 2√7 =
-4√7
10) $-\sqrt{216}$
216 = 36 × 6 → √216 = 6√6
So:
-6√6
11) $-6\sqrt{384}$
384 = 64 × 6 → √384 = 8√6
So: -6 × 8√6 =
-48√6
12) $2\sqrt{96}$
96 = 16 × 6 → √96 = 4√6
So: 2 × 4√6 =
8√6
13) $4\sqrt{384}$
Same as #11 but positive: 4 × 8√6 =
32√6
14) $5\sqrt{80}$
80 = 16 × 5 → √80 = 4√5
So: 5 × 4√5 =
20√5
15) $7\sqrt{245}$
245 = 49 × 5 → √245 = 7√5
So: 7 × 7√5 =
49√5
16) $8\sqrt{36}$
√36 = 6 (perfect square!)
So: 8 × 6 =
48
17) $-8\sqrt{144}$
√144 = 12
So: -8 × 12 =
-96
18) $2\sqrt{16}$
√16 = 4
So: 2 × 4 =
8
All answers verified with prime factorization and perfect square extraction.
Final Answer:
1) -24√3
2) -30√3
3) -10√2
4) 64√2
5) -24√2
6) 14√2
7) 24√2
8) -24√7
9) -4√7
10) -6√6
11) -48√6
12) 8√6
13) 32√6
14) 20√5
15) 49√5
16) 48
17) -96
18) 8
Parent Tip: Review the logic above to help your child master the concept of simplest radical form worksheet.
