7) Simplify: a) √9².8²; b) √3⁴⋅5².10²; c) √6.4⋅10³; d) √12.1⋅10⁵; e) √12.96⋅10⁴; f) √0.1⋅10⁷

Simplification of square roots:

a) Simplify √9².8²:

• Using the property √(a²b²) = ab, we get:
• √9².8² = 9 * 8 = 72.

b) Simplify √3⁴⋅5².10²:

• Using the property √(a²ⁿ) = aⁿ and √(ab) = √a√b:
• √3⁴⋅5²⋅10² = √(3²)² ⋅ 5² ⋅ 10² = 3² ⋅ 5 ⋅ 10 = 9 ⋅ 5 ⋅ 10 = 450.

c) Simplify √6.4⋅10³:

• Rewrite 10³ as 10² ⋅ 10:
• √6.4⋅10³ = √(6.4⋅10⋅10²) = √(64⋅10²) = √64 ⋅ √10² = 8 ⋅ 10 = 80.

d) Simplify √12.1⋅10⁵:

• Rewrite 10⁵ as 10⁴ ⋅ 10:
• √12.1⋅10⁵ = √(12.1⋅10⋅10⁴) = √(121⋅10⁴) = √121 ⋅ √10⁴ = 11 ⋅ 10² = 11 ⋅ 100 = 1100.

e) Simplify √12.96⋅10⁴:

• We know that 12.96 = 3.6²:
• √12.96⋅10⁴ = √(3.6²⋅(10²)²) = 3.6 ⋅ 10² = 3.6 ⋅ 100 = 360.

f) Simplify √0.1⋅10⁷:

• Rewrite 10⁷ as 10⁶ ⋅ 10:
• √0.1⋅10⁷ = √(0.1⋅10⋅10⁶) = √(1⋅10⁶) = √1 ⋅ √10⁶ = 1 ⋅ 10³ = 1000.

Answer: a) 72, b) 450, c) 80, d) 1100, e) 360, f) 1000