Math Problem Statement

What is the product of (3√2)(5√10)?

Solution

We are asked to find the product of (32)(510)(3\sqrt{2})(5\sqrt{10}).

Step-by-Step Solution:

  1. Multiply the whole numbers: 3×5=153 \times 5 = 15.
  2. Multiply the square roots: 2×10=2×10=20\sqrt{2} \times \sqrt{10} = \sqrt{2 \times 10} = \sqrt{20}.

Thus, the product is: (32)(510)=1520(3\sqrt{2})(5\sqrt{10}) = 15\sqrt{20}

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

  2. Multiply this by 15: 1520=15×25=30515\sqrt{20} = 15 \times 2\sqrt{5} = 30\sqrt{5}

So, the final answer is: 30530\sqrt{5}

Would you like further details or have any questions?

Here are five related questions to extend your understanding:

  1. How do you multiply two square roots like a×b\sqrt{a} \times \sqrt{b}?
  2. Can all square roots be simplified? Why or why not?
  3. What is the result of squaring a square root, like (5)2(\sqrt{5})^2?
  4. How do you simplify expressions involving square roots and whole numbers together?
  5. What are some real-world applications where square roots are used?

Tip: When simplifying square roots, always look for perfect squares inside the radical (like 4\sqrt{4} or 9\sqrt{9}) to simplify your expression further.

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Math Problem Analysis

Mathematical Concepts

Algebra
Radicals
Multiplication of Square Roots

Formulas

a√b × c√d = ac√(b × d)
√ab = √a × √b

Theorems

Properties of square roots
Multiplication of radicals

Suitable Grade Level

Grades 9-10