Least Common Denominators: Video e Problemi di Pratica

Which statement best defines the least common denominator (LCD) for rational expressions?

Factor the numerical denominators $84$ and $36$ completely into prime factors, then find the LCD of $\frac{1}{84}$ and $\frac{1}{36}$.

Given denominators $12x^2$ and $18x^3$, which of the following is the least common denominator?

Find the LCD of the denominators $45x$ and $12x^2$.

If the LCD of two denominators is $60x^2$ and the denominators are $30x$ and $20x^2$, what are the missing factor(s) for $30x$ and for $20x^2$ respectively?

Rewrite the pair of rational expressions $\frac{1}{30x}$ and $\frac{1}{20x^2}$ so both have the LCD $60x^2$.

Which of the following denominators is already fully factored and is irreducible over the integer values?

Find the LCD for the denominators: $8x$, $12x^2$, and $18x^3$.

Compute the LCD for the denominators $9x$, $20x^2$, and $14$.

For denominators: $(x+2)(x+5)$ and $(x+5)$, with a LCD of $(x+2)(x+5)$, what are the missing factor(s) that must be multiplied to $(x+5)$ in order to reach the LCD value? Also, what should be multiplied to $(x+2)(x+5)$?

Write the equivalent forms with denominator $(x+2)(x+5)$ for the expressions $\frac{3}{x+5}$ and $\frac{2x}{x+2}$.

Which verification step correctly confirms that the expression $\frac{2x}{60x^2}$ is equivalent to $\frac{1}{30x}$?

You rewrite $\frac{1}{6x}$ as $\frac{x+2}{6x(x+2)}$. A classmate claims that this is not equivalent to $\frac{1}{6x}$. Which of the following statements provides the best justification that the rewrite is correct?

Factor the quadratic denominator $x^2+7x+12$ into irreducible factors over the integers.

Factor $x^2+x-20$ completely over the integers.

Find the sum: $\frac{4}{x^2+7x+10}+\frac{3}{x+5}$

Add the rational expressions: $\frac{2}{x+3}+\frac{x}{x^2+6x+9}$

A student claims that the LCD of the denominators: $8$, $12$, and $18$ will be: $2^3\times 3^2\times 5=360$. Is the student's selected LCD correct? Why or why not?

Which method is appropriate when factoring a quadratic denominator with a leading coefficient of $1$ to prepare for an LCD computation?

A chemist combines two solutions. The amounts of solute, in units of moles, for each solution is given by: $\frac{5}{x+4}$ and $\frac{2}{x^2+8x+16}$. What is the total amount of solute?

Given the denominators: $x(x-1)$ and $x^2-1$, determine what the LCD is and specify which missing factor(s) should be multiplied to $x(x-1)$ in order to obtain the LCD.

Which set of factors correctly lists the unique prime and variable components, with highest exponents, for the denominators: $4x^2$, $6x$, and $9x^3$?

Three students computed the LCD of the denominators: $5x^2$, $18x$, and $12x^3$ as follows:

Student A: $2^2\times 3^2\times 5\times x^3=180x^3$

Student B: $2^2\times 3^2\times 5\times x^2=180x^2$

Student C: $2\times 3^2\times 5\times x^3=90x^3$

Which student is correct? 

Given the expressions: $\frac{x}{x^2-1}$ and $\frac{2}{x+1}$, rewrite both over the LCD. Note that the numerators are: $x$ and $2(x-1)$, respectively. Is the rewrite correct? If not, give the correct rewritten numerators over the LCD.

Which quadratic below is considered to be prime over the integers and therefore should be kept as-is when forming an LCD?

Provide the LCD and the rewritten numerators for the following rational expressions: $\frac{1}{x+2}$, $\frac{2}{x^2+3x+2}$, and $\frac{3}{2(x+1)}$.

After rewriting two expressions over the LCD, a student gets the numerators: $5x+3$ and $10x+6$. The student cancels a common factor and claims that the two rewritten numerators are proportional. Which of the following justifications correctly explains whether or not the numerators were scaled appropriately?

If the LCD of two denominators is $84$ and one denominator is $28$, what must be multiplied to $28$ to reach the LCD?

For the denominators: $16x^2$, $24x^3$, and $9x$, a student claims that the LCD is: $2^4\cdot 3^2\cdot x^3=16\cdot 9\cdot x^3=144x^3$. Is this the correct least common denominator? Justify your conclusion.

Two painters are working together to paint a wall. Painter A can paint $\frac{3}{x+2}$ of the wall per hour, while Painter B can paint $\frac{2}{x^2+3x+2}$ of the wall per hour. Determine their combined rate.