Simplify square root of (3x)/(23y)
The question is asking for the simplification of the expression containing a square root of a fraction. Specifically, the expression inside the square root is the fraction formed by the product of 3 and a variable x in the numerator and the product of 23 and another variable y in the denominator. The task is to rewrite this square root expression in a simpler or more reduced form, where possible, by applying the properties of square roots and fractions.
$\sqrt{\frac{3 x}{23 y}}$
Answer
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: $\frac{\sqrt{3x}\sqrt{23y}}{(23y)^{\frac{1}{2}\cdot2}}$.
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: $\frac{\sqrt{3x\cdot23y}}{23y}$.
Step 4.2:
Perform the multiplication inside the radical: $\frac{\sqrt{69xy}}{23y}$.
Knowledge Notes:
Square Root of a Quotient: The square root of a quotient can be expressed as a quotient of square roots, $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.
Rationalizing the Denominator: Involves multiplying the numerator and denominator by a conjugate or a suitable term to eliminate the square root from the denominator.
Exponent Rules:
Power of a Power: $(a^m)^n = a^{mn}$.
Product of Powers: $a^m \cdot a^n = a^{m+n}$.
Simplifying Square Roots: $\sqrt{a^2} = a$, assuming $a$ is non-negative.
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.
Simplifying Expressions: Involves combining like terms, applying exponent rules, and performing arithmetic operations to express the result in its simplest form.
