Math Problem Statement

Simplify. Assume a is greater than or equal to zero. \( \sqrt{18a^5} \)

Solution

The given expression is:

18a5\sqrt{18a^5}

We are asked to simplify this expression, assuming a0a \geq 0.

Step-by-step simplification:

  1. Factor the radicand: 18a5=18×a5\sqrt{18a^5} = \sqrt{18} \times \sqrt{a^5} We can simplify 18\sqrt{18}. Since 18=9×218 = 9 \times 2, we have: 18=9×2=9×2=32\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}

  2. Simplify a5\sqrt{a^5}: We break a5a^5 into a4×aa^4 \times a, where a4a^4 is a perfect square: a5=a4×a=a4×a=a2×a\sqrt{a^5} = \sqrt{a^4 \times a} = \sqrt{a^4} \times \sqrt{a} = a^2 \times \sqrt{a}

  3. Combine the results: Now, substituting back into the original expression: 18a5=32×a2×a\sqrt{18a^5} = 3\sqrt{2} \times a^2 \times \sqrt{a} This simplifies to: 3a22a3a^2\sqrt{2a}

Thus, the simplified expression is: 3a22a3a^2\sqrt{2a}

Would you like any further details or clarification?

Related Questions:

  1. How do you simplify square roots involving both constants and variables?
  2. What is the process for factoring numbers under a square root?
  3. How does the assumption a0a \geq 0 affect the solution?
  4. Can this expression be further simplified if we know specific values for aa?
  5. How does the exponent rule apply to square roots?

Tip:

Always look for perfect squares when simplifying square roots—they can greatly reduce the complexity of the expression.

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Math Problem Analysis

Mathematical Concepts

Simplification
Square Roots
Exponents
Radical Expressions

Formulas

√(x * y) = √x * √y
√(a^n) = a^(n/2)

Theorems

Product Rule of Radicals
Exponent Rules

Suitable Grade Level

Grades 9-11