Math Problem Statement

Express in simplest radical form: 2x^3 * sqrt(180) + x^2 * sqrt(45x^2).

Solution

To simplify the given expression 2x3180+x245x22x^3 \sqrt{180} + x^2 \sqrt{45x^2}, let's work through it step-by-step:

  1. Simplify each radical term:

    • For the first term, 2x31802x^3 \sqrt{180}: 180=365=65\sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5} So, 2x3180=2x365=12x352x^3 \sqrt{180} = 2x^3 \cdot 6\sqrt{5} = 12x^3 \sqrt{5}.

    • For the second term, x245x2x^2 \sqrt{45x^2}: 45x2=95x2=3x5\sqrt{45x^2} = \sqrt{9 \cdot 5 \cdot x^2} = 3x \sqrt{5} Therefore, x245x2=x23x5=3x35x^2 \sqrt{45x^2} = x^2 \cdot 3x \sqrt{5} = 3x^3 \sqrt{5}.

  2. Combine like terms:

    Now the expression becomes: 12x35+3x3512x^3 \sqrt{5} + 3x^3 \sqrt{5} Factor out x35x^3 \sqrt{5}: =(12+3)x35=15x35= (12 + 3)x^3 \sqrt{5} = 15x^3 \sqrt{5}

Thus, the expression in simplest radical form is: 15x35\boxed{15x^3 \sqrt{5}}

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


Here are some related questions you might explore:

  1. How do you simplify radicals when the radicand is a product of squares?
  2. What is the process to factor out like terms in an expression?
  3. How does the distributive property work in simplifying expressions with radicals?
  4. Can you apply these steps to simplify a different radical expression?
  5. What are some other properties of square roots that help with simplification?

Tip: Always look for perfect squares within the radicand to simplify radicals more easily.

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Math Problem Analysis

Mathematical Concepts

Algebra
Radicals
Simplifying Expressions

Formulas

sqrt(a*b) = sqrt(a) * sqrt(b)
Combining like terms

Theorems

Properties of Square Roots
Distributive Property

Suitable Grade Level

Grades 10-12