Simplify square root of (3x)/(23y)

Solution:

Simplification of $\sqrt{\frac{3x}{23y}}$

Step 1:

Express $\sqrt{\frac{3x}{23y}}$ as a quotient of square roots: $\frac{\sqrt{3x}}{\sqrt{23y}}$.

Step 2:

Rationalize the denominator by multiplying the fraction by $\frac{\sqrt{23y}}{\sqrt{23y}}$.

Step 3:

Simplify the expression in the denominator.

Step 3.1:

Combine the numerator and denominator: $\frac{\sqrt{3x}\sqrt{23y}}{\sqrt{23y}\sqrt{23y}}$.

Step 3.2:

Express the denominator as a power: $\frac{\sqrt{3x}\sqrt{23y}}{(\sqrt{23y}{)}^1\sqrt{23y}}$.

Step 3.3:

Repeat the expression of the denominator as a power: $\frac{\sqrt{3x}\sqrt{23y}}{(\sqrt{23y}{)}^1(\sqrt{23y}{)}^1}$.

Step 3.4:

Apply the exponent rule: $\frac{\sqrt{3x}\sqrt{23y}}{(\sqrt{23y}{)}^{1+1}}$.

Step 3.5:

Add the exponents: $\frac{\sqrt{3x}\sqrt{23y}}{(\sqrt{23y}{)}^2}$.

Step 3.6:

Simplify the square of a square root in the denominator.

Step 3.6.1:

Represent the square root as a power: $\frac{\sqrt{3x}\sqrt{23y}}{((23y{)}^{\frac{1}{2}}{)}^2}$.

Step 3.6.2:

Apply the power of a power rule: $\frac{\sqrt{3x}\sqrt{23y}}{(23y{)}^{\frac{1}{2}\cdot 2}}$.

Step 3.6.3:

Multiply the exponents: $\frac{\sqrt{3x}\sqrt{23y}}{(23y{)}^{\frac{2}{2}}}$.

Step 3.6.4:

Simplify the fraction of exponents.

Step 3.6.4.1:

Cancel out the common factors: $\frac{\sqrt{3x}\sqrt{23y}}{(23y{)}^{\frac{\cancel{2}}{\cancel{2}}}}$.

Step 3.6.4.2:

Finalize the denominator: $\frac{\sqrt{3x}\sqrt{23y}}{(23y{)}^1}$.

Step 3.6.5:

Simplify the expression: $\frac{\sqrt{3x}\sqrt{23y}}{23y}$.

Step 4:

Simplify the numerator.

Step 4.1:

Combine under a single radical: $\frac{\sqrt{3x\cdot 23y}}{23y}$.

Step 4.2:

Perform the multiplication inside the radical: $\frac{\sqrt{69xy}}{23y}$.

Knowledge Notes:

1. Square Root of a Quotient: The square root of a quotient can be expressed as a quotient of square roots, $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$.

2. Rationalizing the Denominator: Involves multiplying the numerator and denominator by a conjugate or a suitable term to eliminate the square root from the denominator.

3. Exponent Rules:

   • Power of a Power: $(a^m{)}^n=a^{mn}$.

   • Product of Powers: $a^m\cdot a^n=a^{m+n}$.

4. Simplifying Square Roots: $\sqrt{a^2}=a$, assuming $a$ is non-negative.

5. Multiplication Inside a Radical: The product rule for radicals states that $\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}$, provided $a$ and $b$ are non-negative.

6. Simplifying Expressions: Involves combining like terms, applying exponent rules, and performing arithmetic operations to express the result in its simplest form.