Math Problem Statement

Simplify each expression as presented in the image: a) 5√20 b) √28 + 3√7 c) 3√8 - 2√18 d) 2√7 × 3√2 e) 2√5 × √20 f) 15√12 ÷ 5√3 g) √3(√3 + √15) h) (√7 + √2)^2 i) (√5 + √6)(√2 + √5)

Solution

Let's simplify each expression step-by-step:

a) 5205\sqrt{20}

Simplify 20\sqrt{20}: 20=4×5=25\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}

So, the expression becomes: 5×25=1055 \times 2\sqrt{5} = 10\sqrt{5}

b) 28+37\sqrt{28} + 3\sqrt{7}

Simplify 28\sqrt{28}: 28=4×7=27\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}

So, the expression becomes: 27+37=572\sqrt{7} + 3\sqrt{7} = 5\sqrt{7}

c) 382183\sqrt{8} - 2\sqrt{18}

Simplify 8\sqrt{8} and 18\sqrt{18}: 8=4×2=22and18=9×2=32\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \quad \text{and} \quad \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}

So, the expression becomes: 3×222×32=6262=03 \times 2\sqrt{2} - 2 \times 3\sqrt{2} = 6\sqrt{2} - 6\sqrt{2} = 0

d) 27×322\sqrt{7} \times 3\sqrt{2}

Multiply the terms: (2×3)×(7×2)=614(2 \times 3) \times (\sqrt{7} \times \sqrt{2}) = 6\sqrt{14}

e) 25×202\sqrt{5} \times \sqrt{20}

Simplify 20\sqrt{20}: 20=25\sqrt{20} = 2\sqrt{5}

Multiply the terms: 25×25=4×5=202\sqrt{5} \times 2\sqrt{5} = 4 \times 5 = 20

f) 1512÷5315\sqrt{12} \div 5\sqrt{3}

Simplify 12\sqrt{12}: 12=4×3=23\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}

Divide the terms: 15×2353=30353=6\frac{15 \times 2\sqrt{3}}{5\sqrt{3}} = \frac{30\sqrt{3}}{5\sqrt{3}} = 6

g) 3(3+15)\sqrt{3}(\sqrt{3} + \sqrt{15})

Distribute 3\sqrt{3}: 3×3+3×15=3+45\sqrt{3} \times \sqrt{3} + \sqrt{3} \times \sqrt{15} = 3 + \sqrt{45}

Simplify 45\sqrt{45}: 45=9×5=35\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}

So, the expression becomes: 3+353 + 3\sqrt{5}

h) (7+2)2(\sqrt{7} + \sqrt{2})^2

Expand using the square of a binomial: (7+2)2=(7)2+2(7)(2)+(2)2=7+214+2=9+214(\sqrt{7} + \sqrt{2})^2 = (\sqrt{7})^2 + 2(\sqrt{7})(\sqrt{2}) + (\sqrt{2})^2 = 7 + 2\sqrt{14} + 2 = 9 + 2\sqrt{14}

i) (5+6)(2+5)(\sqrt{5} + \sqrt{6})(\sqrt{2} + \sqrt{5})

Expand using FOIL (First, Outside, Inside, Last): 5×2+5×5+6×2+6×5=10+5+12+30\sqrt{5} \times \sqrt{2} + \sqrt{5} \times \sqrt{5} + \sqrt{6} \times \sqrt{2} + \sqrt{6} \times \sqrt{5} = \sqrt{10} + 5 + \sqrt{12} + \sqrt{30}

Simplify 12\sqrt{12}: 12=23\sqrt{12} = 2\sqrt{3}

So, the expression becomes: 10+5+23+30\sqrt{10} + 5 + 2\sqrt{3} + \sqrt{30}

Would you like more details on any specific problem, or do you have any questions?

Related Questions:

  1. How do you simplify expressions with multiple radicals?
  2. What are the rules for multiplying and dividing square roots?
  3. How can you simplify expressions with binomials involving radicals?
  4. What are common mistakes when working with radicals?
  5. How do you expand and simplify expressions using the distributive property?

Tip: Always simplify the radicals first before performing other operations; this often makes the calculations easier!

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Math Problem Analysis

Mathematical Concepts

Radicals
Square Roots
Simplification
Multiplication of Radicals
Binomial Expansion

Formulas

√(a × b) = √a × √b
(a + b)^2 = a^2 + 2ab + b^2
(a + b)(c + d) = ac + ad + bc + bd

Theorems

Properties of square roots
Distributive property
Simplifying radicals

Suitable Grade Level

Grades 9-11