Simplify square root of (225x^3)/(49x)
The question asks to perform the mathematical operation of simplification on a given expression. The expression involves a square root of a fraction. The fraction consists of two parts: the numerator, which is 225 times x to the power of 3, and the denominator, which is 49 times x. The task is to simplify this complex square root by breaking it down into simpler components, factoring out perfect squares where possible, and canceling terms if applicable to simplify the expression to its simplest form.
$\sqrt{\frac{225 x^{3}}{49 x}}$
Answer
Solution:
Step 1
Simplify the fraction $\frac{225x^3}{49x}$ by removing common terms.
Step 1.1
Extract $x$ from $225x^3$ to get $\sqrt{\frac{x(225x^2)}{49x}}$.
Step 1.2
Extract $x$ from $49x$ to get $\sqrt{\frac{x(225x^2)}{x \cdot 49}}$.
Step 1.3
Eliminate the common $x$ term to get $\sqrt{\frac{\cancel{x}(225x^2)}{\cancel{x} \cdot 49}}$.
Step 1.4
Reformulate the expression as $\sqrt{\frac{225x^2}{49}}$.
Step 2
Express $225x^2$ as $(15x)^2$ to get $\sqrt{\frac{(15x)^2}{49}}$.
Step 3
Represent $49$ as $7^2$ to get $\sqrt{\frac{(15x)^2}{7^2}}$.
Step 4
Convert $\frac{(15x)^2}{7^2}$ to $(\frac{15x}{7})^2$ to get $\sqrt{(\frac{15x}{7})^2}$.
Step 5
Extract terms from under the square root, assuming all are positive real numbers, to obtain $\frac{15x}{7}$.
Knowledge Notes:
The problem involves simplifying a square root of a rational expression. The process includes several algebraic techniques:
Factorization: Breaking down expressions into products of factors, which can reveal common terms that may cancel out.
Cancellation: When a term appears in both the numerator and the denominator of a fraction, it can be cancelled out, simplifying the expression.
Square Roots and Exponents: Understanding that $\sqrt{x^2} = x$ for any positive real number $x$, and that $\sqrt{\frac{a^2}{b^2}} = \frac{a}{b}$ for any positive real numbers $a$ and $b$.
Rationalizing the Denominator: In some cases, it might be necessary to manipulate the expression to remove square roots from the denominator, but this is not required in this problem.
Assumption of Positive Real Numbers: When simplifying square roots, we typically assume that variables represent positive real numbers to avoid dealing with complex numbers or absolute values.
In this problem, we use these techniques to simplify the given square root expression step by step. The process involves factoring out common terms, rewriting squared terms, and finally extracting the square root of a perfect square.
