Math Problem Statement

Solve the following radical expressions: 1. 3√7 + √700 - √28 2. √125 + √720 - 9√5 3. √400 + √125 + √500 4. √48 - √768 + 7√3 5. 9√2 - √578 + √72

Solution

Let's solve each of the problems step by step.

1. 37+700283\sqrt{7} + \sqrt{700} - \sqrt{28}

  • Simplify the square roots: 700=100×7=107,28=4×7=27\sqrt{700} = \sqrt{100 \times 7} = 10\sqrt{7}, \quad \sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}
  • Substitute and combine like terms: 37+10727=(3+102)7=1173\sqrt{7} + 10\sqrt{7} - 2\sqrt{7} = (3 + 10 - 2)\sqrt{7} = 11\sqrt{7}
  • Answer: 11711\sqrt{7}

2. 125+72095\sqrt{125} + \sqrt{720} - 9\sqrt{5}

  • Simplify the square roots: 125=25×5=55,720=144×5=125\sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5}, \quad \sqrt{720} = \sqrt{144 \times 5} = 12\sqrt{5}
  • Substitute and combine like terms: 55+12595=(5+129)5=855\sqrt{5} + 12\sqrt{5} - 9\sqrt{5} = (5 + 12 - 9)\sqrt{5} = 8\sqrt{5}
  • Answer: 858\sqrt{5}

3. 400+125+500\sqrt{400} + \sqrt{125} + \sqrt{500}

  • Simplify the square roots: 400=20,125=55,500=100×5=105\sqrt{400} = 20, \quad \sqrt{125} = 5\sqrt{5}, \quad \sqrt{500} = \sqrt{100 \times 5} = 10\sqrt{5}
  • Combine like terms: 20+55+105=20+15520 + 5\sqrt{5} + 10\sqrt{5} = 20 + 15\sqrt{5}
  • Answer: 20+15520 + 15\sqrt{5}

4. 48768+73\sqrt{48} - \sqrt{768} + 7\sqrt{3}

  • Simplify the square roots: 48=16×3=43,768=256×3=163\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}, \quad \sqrt{768} = \sqrt{256 \times 3} = 16\sqrt{3}
  • Substitute and combine like terms: 43163+73=(416+7)3=534\sqrt{3} - 16\sqrt{3} + 7\sqrt{3} = (4 - 16 + 7)\sqrt{3} = -5\sqrt{3}
  • Answer: 53-5\sqrt{3}

5. 92578+729\sqrt{2} - \sqrt{578} + \sqrt{72}

  • Simplify the square roots: 72=36×2=62,578 is not a simple square root.\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}, \quad \sqrt{578} \text{ is not a simple square root}.
  • Approximate 57824.04\sqrt{578} \approx 24.04, and simplify the expression: 92+6224.04=15224.049\sqrt{2} + 6\sqrt{2} - 24.04 = 15\sqrt{2} - 24.04
  • Answer: 15224.0415\sqrt{2} - 24.04

Would you like more details or have any questions about these solutions?

Here are 5 related questions:

  1. How do you simplify square roots of non-perfect squares?
  2. Can you explain how to combine like terms in radical expressions?
  3. How do you approximate square roots without a calculator?
  4. How does factoring help in simplifying square roots?
  5. What are some common square root simplifications?

Tip: Always look for factors of the radicand (the number under the square root) that are perfect squares to simplify square roots efficiently!

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Math Problem Analysis

Mathematical Concepts

Radical Expressions
Simplifying Square Roots
Combining Like Terms

Formulas

√a × b = √a × √b
a√n ± b√n = (a ± b)√n

Theorems

Simplification of Square Roots
Properties of Radicals

Suitable Grade Level

Grades 9-12