Math Problem Statement

Simplify \( \sqrt{9x^{13}} \) assuming that the variable represents a positive real number.

Solution

The expression you provided is:

9x13\sqrt{9x^{13}}

We need to simplify this under the assumption that xx is a positive real number.

Step-by-step simplification:

  1. Factor the expression inside the square root: 9x13=9x13\sqrt{9x^{13}} = \sqrt{9} \cdot \sqrt{x^{13}}

  2. Simplify 9\sqrt{9}: 9=3\sqrt{9} = 3

  3. Simplify x13\sqrt{x^{13}}: Use the property of square roots: xa=xa/2\sqrt{x^a} = x^{a/2}. So, x13=x13/2\sqrt{x^{13}} = x^{13/2}

  4. Combine the results: 3x13/23 \cdot x^{13/2}

Thus, the simplified expression is:

3x13/23x^{13/2}

Would you like more details, or do you have any questions on how this was done?

Relative questions:

  1. What is the rule for simplifying square roots of powers, and why does x13x^{13} become x13/2x^{13/2}?
  2. Can you explain why 9\sqrt{9} equals 3?
  3. How would the answer change if xx were a negative number?
  4. How can this simplification be used in solving equations or practical applications?
  5. What is the difference between simplifying expressions and solving equations?

Tip:

When simplifying square roots involving exponents, always remember to halve the exponent of any variable under the radical.

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Math Problem Analysis

Mathematical Concepts

Simplification
Square Roots
Exponents
Radicals

Formulas

\( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)
\( \sqrt{x^a} = x^{a/2} \)

Theorems

Properties of Exponents
Properties of Square Roots

Suitable Grade Level

Grades 9-12