Simplify (3y)/( square root of 9x^3y^4)

Solution:

Step 1: Simplify the denominator

• Step 1.1: Express $9x^3y^4$ as $(3xy^2)^2 \cdot x$.

  • Step 1.1.1: Write $9$ as $3^2$. $\frac{3y}{\sqrt{3^2x^3y^4}}$
  • Step 1.1.2: Extract $x^2$. $\frac{3y}{\sqrt{3^2(x^2 \cdot x)y^4}}$
  • Step 1.1.3: Represent $y^4$ as $(y^2)^2$. $\frac{3y}{\sqrt{3^2(x^2 \cdot x)(y^2)^2}}$
  • Step 1.1.4: Rearrange $x$. $\frac{3y}{\sqrt{3^2(x^2)(y^2)^2 \cdot x}}$
  • Step 1.1.5: Combine as $(3xy^2)^2$. $\frac{3y}{\sqrt{(3xy^2)^2 \cdot x}}$

• Step 1.2: Extract terms from the radical. $\frac{3y}{3xy^2\sqrt{x}}$

Step 2: Reduce the expression by cancelling common factors

• Step 2.1: Eliminate the common factor of $3$.

  • Step 2.1.1: Remove the common factor. $\frac{\cancel{3}y}{\cancel{3}xy^2\sqrt{x}}$
  • Step 2.1.2: Simplify the expression. $\frac{y}{xy^2\sqrt{x}}$

• Step 2.2: Cancel the common factors of $y$ and $y^2$.

  • Step 2.2.1: Elevate $y$ to the power of $1$. $\frac{y^1}{xy^2\sqrt{x}}$

  • Step 2.2.2: Separate $y$ from $y^1$. $\frac{y \cdot 1}{xy^2\sqrt{x}}$

  • Step 2.2.3: Remove common factors.

    • Step 2.2.3.1: Extract $y$ from $xy^2\sqrt{x}$. $\frac{y \cdot 1}{y(x y \sqrt{x})}$
    • Step 2.2.3.2: Cancel the common factor. $\frac{\cancel{y} \cdot 1}{\cancel{y}(xy\sqrt{x})}$
    • Step 2.2.3.3: Present the simplified expression. $\frac{1}{xy\sqrt{x}}$

Step 3: Rationalize the denominator by multiplying by $\frac{\sqrt{x}}{\sqrt{x}}$. $\frac{1}{xy\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 4: Combine and simplify the denominator

• Step 4.1: Multiply by $\frac{\sqrt{x}}{\sqrt{x}}$. $\frac{\sqrt{x}}{xy\sqrt{x}\sqrt{x}}$

• Step 4.2: Rearrange $\sqrt{x}$. $\frac{\sqrt{x}}{xy(\sqrt{x}\sqrt{x})}$

• Step 4.3: Elevate $\sqrt{x}$ to the power of $1$. $\frac{\sqrt{x}}{xy((\sqrt{x})^1\sqrt{x})}$

• Step 4.4: Apply the same elevation. $\frac{\sqrt{x}}{xy((\sqrt{x})^1(\sqrt{x})^1)}$

• Step 4.5: Use the power rule $a^m a^n = a^{m+n}$ to combine exponents. $\frac{\sqrt{x}}{xy(\sqrt{x})^{1+1}}$

• Step 4.6: Add $1$ and $1$. $\frac{\sqrt{x}}{xy(\sqrt{x})^2}$

• Step 4.7: Convert $(\sqrt{x})^2$ to $x$.

  • Step 4.7.1: Rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$. $\frac{\sqrt{x}}{xy(x^{\frac{1}{2}})^2}$

  • Step 4.7.2: Apply the power rule, $(a^m)^n = a^{mn}$. $\frac{\sqrt{x}}{xyx^{\frac{1}{2} \cdot 2}}$

  • Step 4.7.3: Simplify $\frac{1}{2} \cdot 2$. $\frac{\sqrt{x}}{xyx^{\frac{2}{2}}}$

  • Step 4.7.4: Remove the common factor of $2$.

    • Step 4.7.4.1: Cancel the common factor. $\frac{\sqrt{x}}{xyx^{\frac{\cancel{2}}{\cancel{2}}}}$
    • Step 4.7.4.2: Display the simplified expression. $\frac{\sqrt{x}}{xyx^1}$

  • Step 4.7.5: Simplify further. $\frac{\sqrt{x}}{xyx}$

Step 5: Combine $x$ terms by adding exponents

• Step 5.1: Rearrange $x$. $\frac{\sqrt{x}}{x \cdot x y}$
• Step 5.2: Multiply $x$ by $x$. $\frac{\sqrt{x}}{x^2 y}$

Knowledge Notes:

• Radical Simplification: To simplify a radical, factors inside the radical can be rewritten such that perfect squares are extracted, reducing the complexity of the expression.

• Rationalizing the Denominator: When a radical is present in the denominator, it is common practice to multiply the fraction by a form of 1 that will eliminate the radical from the denominator. This process is known as rationalization.

• Common Factor Cancellation: When the same factor appears in both the numerator and the denominator, it can be cancelled out to simplify the fraction.

• Exponent Rules:

  • Power Rule: $a^m a^n = a^{m+n}$ allows us to combine bases with exponents by adding the exponents.

  • Power of a Power Rule: $(a^m)^n = a^{mn}$ allows us to multiply exponents when an exponent is raised to another power.

  • Square Root as an Exponent: $\sqrt[n]{a^x} = a^{\frac{x}{n}}$ allows us to express a root as a fractional exponent.

• Simplifying Expressions: The goal of simplification is to write the expression in the simplest form by combining like terms, reducing fractions, and applying algebraic rules.