Math Problem Statement

(3x)^2* (1/9x)1/2

Solution

Let's solve the expression step by step:

(3x)2×(19x)12(3x)^2 \times \left( \frac{1}{9x} \right)^{\frac{1}{2}}

Step 1: Simplify (3x)2(3x)^2

(3x)2=9x2(3x)^2 = 9x^2

Step 2: Simplify (19x)12\left( \frac{1}{9x} \right)^{\frac{1}{2}}

(19x)12=19x=13x\left( \frac{1}{9x} \right)^{\frac{1}{2}} = \frac{1}{\sqrt{9x}} = \frac{1}{3\sqrt{x}}

Step 3: Multiply the two expressions

9x2×13x=9x23x=3x2÷x9x^2 \times \frac{1}{3\sqrt{x}} = \frac{9x^2}{3\sqrt{x}} = 3x^{2} \div \sqrt{x}

Step 4: Simplify x2x\frac{x^2}{\sqrt{x}}

x2x=x212=x32\frac{x^2}{\sqrt{x}} = x^{2 - \frac{1}{2}} = x^{\frac{3}{2}}

Final expression:

3x323x^{\frac{3}{2}}

So, the simplified form of the expression is: 3x323x^{\frac{3}{2}}

Would you like further clarification or more details on any of these steps?


Here are 5 related questions to deepen your understanding:

  1. How do you simplify expressions with fractional exponents?
  2. What is the process for multiplying terms with exponents?
  3. How do you simplify radical expressions involving variables?
  4. How does raising a fraction to a power work in different cases?
  5. What is the relationship between exponents and roots?

Tip: When simplifying expressions with exponents, always pay attention to the exponent rules, especially when handling radicals and fractional exponents.

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponentiation
Radicals
Simplification

Formulas

(a^m)^n = a^{m*n}
Radical to Exponent Conversion: √x = x^(1/2)
Multiplication of Exponents: a^m * a^n = a^{m+n}

Theorems

Exponent Rules
Radical Simplification

Suitable Grade Level

Grades 9-11