Simplify (x square root of 18x^3)/3

Solution:

Step 1: Simplify the numerator.

• Step 1.1: Express $18x^3$ as $(3x{)}^2\cdot 2x$.

• Step 1.1.1: Extract $9$ from $18$ to get $\frac{x\sqrt{9\cdot 2x^3}}{3}$.

• Step 1.1.2: Represent $9$ as $3^2$ to obtain $\frac{x\sqrt{3^2\cdot 2x^3}}{3}$.

• Step 1.1.3: Separate $x^2$ from $x^3$ to rewrite as $\frac{x\sqrt{3^2\cdot 2(x^2\cdot x)}}{3}$.

• Step 1.1.4: Rearrange to $\frac{x\sqrt{3^2{x}^2\cdot 2x}}{3}$.

• Step 1.1.5: Rewrite $3^2{x}^2$ as $(3x{)}^2$ to get $\frac{x\sqrt{(3x{)}^2\cdot 2x}}{3}$.

• Step 1.1.6: Enclose $2x$ in parentheses to have $\frac{x\sqrt{(3x{)}^2\cdot (2x)}}{3}$.

• Step 1.2: Extract terms from under the square root as $\frac{x\cdot 3x\sqrt{2x}}{3}$.

• Step 1.3: Combine the exponents.

  • Step 1.3.1: Elevate $x$ to the power of $1$ to form $\frac{3(x^1\cdot x)\sqrt{2x}}{3}$.

  • Step 1.3.2: Again, consider $x$ raised to the power of $1$ as $\frac{3(x^1\cdot x^1)\sqrt{2x}}{3}$.

  • Step 1.3.3: Apply the power rule $a^m\cdot a^n=a^{m+n}$ to combine exponents into $\frac{3x^{1+1}\sqrt{2x}}{3}$.

  • Step 1.3.4: Sum the exponents $1$ and $1$ to get $\frac{3x^2\sqrt{2x}}{3}$.

Step 2: Eliminate the common factor of $3$.

• Step 2.1: Remove the common factor to simplify as $\frac{\cancel{3}{x}^2\sqrt{2x}}{\cancel{3}}$.

• Step 2.2: Divide $x^2\sqrt{2x}$ by $1$ to obtain the final result $x^2\sqrt{2x}$.

Knowledge Notes:

To simplify the expression $(x\sqrt{18x^3})/3$, we need to apply several algebraic rules and properties:

1. Factorization: Breaking down numbers into their prime factors or rewriting expressions in a way that reveals common factors.

2. Square Root Simplification: Recognizing that $\sqrt{a^2}=a$ and applying this to simplify square roots.

3. Exponent Rules: Specifically, the power rule which states that $a^m\cdot a^n=a^{m+n}$.

4. Rational Expression Simplification: Canceling common factors in the numerator and denominator of a fraction.

5. Radical Operations: Understanding how to manipulate expressions under a radical, including factoring out perfect squares to simplify the radical.

By applying these principles, we can simplify the given algebraic expression step by step, ensuring that each transformation is mathematically valid and leads to a simpler form.