Simplify the following algebraic fractions: 1) 3/5x2 - 3/2x, 2) 1/y2 - 3/2y − 1/3y, 3) 3/4t2 - 5/6t2 + 2/3t…

Question

Simplify the following algebraic fractions: 1) 3/5x2 - 3/2x, 2) 1/y2 - 3/2y − 1/3y, 3) 3/4t2 - 5/6t2 + 2/3t, 4) 6x2/(x2-y2) − 3x/(x+y) + 3y/(y-x)

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems.

:

\frac{3}{5x^2} - \frac{3}{2x}

Find the least common multiple (LCM) of the denominators $5x^2$ and $2x$ .
The LCM is $10x^2$ .
Rewrite each fraction with the common denominator.
$\frac{3}{5x^2} \cdot \frac{2}{2} - \frac{3}{2x} \cdot \frac{5x}{5x}$ $\frac{6}{10x^2} - \frac{15x}{10x^2}$
Combine the numerators over the common denominator.
$\frac{6 - 15x}{10x^2}$
The simplified expression is $\boxed{\frac{6 - 15x}{10x^2}}$ .

:
$\frac{1}{y^2} - \frac{3}{2y} - \frac{1}{3y}$
Find the LCM of the denominators $y^2$ , $2y$ , and $3y$ .
The LCM is $6y^2$ .
Rewrite each fraction with the common denominator.
$\frac{1}{y^2} \cdot \frac{6}{6} - \frac{3}{2y} \cdot \frac{3y}{3y} - \frac{1}{3y} \cdot \frac{2y}{2y}$ $\frac{6}{6y^2} - \frac{9y}{6y^2} - \frac{2y}{6y^2}$
Combine the numerators over the common denominator.
$\frac{6 - 9y - 2y}{6y^2}$
Simplify the numerator.
$\frac{6 - 11y}{6y^2}$
The simplified expression is $\boxed{\frac{6 - 11y}{6y^2}}$ .

:
$\frac{3}{4t^2} - \frac{5}{6t^2} + \frac{2}{3t}$
Find the LCM of the denominators $4t^2$ , $6t^2$ , and $3t$ .
The LCM is $12t^2$ .
Rewrite each fraction with the common denominator.
$\frac{3}{4t^2} \cdot \frac{3}{3} - \frac{5}{6t^2} \cdot \frac{2}{2} + \frac{2}{3t} \cdot \frac{4t}{4t}$ $\frac{9}{12t^2} - \frac{10}{12t^2} + \frac{8t}{12t^2}$
Combine the numerators over the common denominator.
$\frac{9 - 10 + 8t}{12t^2}$
Simplify the numerator.
$\frac{-1 + 8t}{12t^2}$
The simplified expression is $\boxed{\frac{8t - 1}{12t^2}}$ .

:
$\frac{6x^2}{x^2-y^2} - \frac{3x}{x+y} + \frac{3y}{y-x}$
Factor the denominators.
Note that $x^2-y^2 = (x-y)(x+y)$ and $y-x = -(x-y)$ .
$\frac{6x^2}{(x-y)(x+y)} - \frac{3x}{x+y} + \frac{3y}{-(x-y)}$ $\frac{6x^2}{(x-y)(x+y)} - \frac{3x}{x+y} - \frac{3y}{x-y}$
Find the LCM of the denominators $(x-y)(x+y)$ , $x+y$ , and $x-y$ .
The LCM is $(x-y)(x+y)$ .
Rewrite each fraction with the common denominator.
$\frac{6x^2}{(x-y)(x+y)} - \frac{3x}{x+y} \cdot \frac{x-y}{x-y} - \frac{3y}{x-y} \cdot \frac{x+y}{x+y}$ $\frac{6x^2}{(x-y)(x+y)} - \frac{3x(x-y)}{(x-y)(x+y)} - \frac{3y(x+y)}{(x-y)(x+y)}$
Combine the numerators over the common denominator.
$\frac{6x^2 - 3x(x-y) - 3y(x+y)}{(x-y)(x+y)}$
Expand the terms in the numerator.
$\frac{6x^2 - (3x^2 - 3xy) - (3xy + 3y^2)}{(x-y)(x+y)}$ $\frac{6x^2 - 3x^2 + 3xy - 3xy - 3y^2}{(x-y)(x+y)}$
Simplify the numerator by combining like terms.
$\frac{3x^2 - 3y^2}{(x-y)(x+y)}$
Factor the numerator and simplify the expression.
$\frac{3(x^2 - y^2)}{(x-y)(x+y)}$ $\frac{3(x-y)(x+y)}{(x-y)(x+y)}$ $3$
The simplified expression is $\boxed{3}$ .

Simplify the following algebraic fractions: 1) 3/5x2 - 3/2x, 2) 1/y2 - 3/2y - 1/3y, 3) 3/4t2 - 5/6t2 + 2/3t, 4) 6x2/(x2-y2) - 3x/(x+y) + 3y/(y-x)

Step-by-step solution

Find the least common multiple (LCM) of the denominators $5x^2$ and $2x$ .

Rewrite each fraction with the common denominator.

Combine the numerators over the common denominator.

Find the LCM of the denominators $y^2$ , $2y$ , and $3y$ .

Rewrite each fraction with the common denominator.

Combine the numerators over the common denominator.

Simplify the numerator.

Find the LCM of the denominators $4t^2$ , $6t^2$ , and $3t$ .

Rewrite each fraction with the common denominator.

Combine the numerators over the common denominator.

Simplify the numerator.

Factor the denominators.

Find the LCM of the denominators $(x-y)(x+y)$ , $x+y$ , and $x-y$ .

Rewrite each fraction with the common denominator.

Combine the numerators over the common denominator.

Expand the terms in the numerator.

Simplify the numerator by combining like terms.

Factor the numerator and simplify the expression.

Still stuck on this one?

More Mathematics Questions

Simplify the following algebraic fractions: 1) 3/5x2 - 3/2x, 2) 1/y2 - 3/2y - 1/3y, 3) 3/4t2 - 5/6t2 + 2/3t, 4) 6x2/(x2-y2) - 3x/(x+y) + 3y/(y-x)

Step-by-step solution

Find the least common multiple (LCM) of the denominators 5x25x^25x2 and 2x2x2x.

Rewrite each fraction with the common denominator.

Combine the numerators over the common denominator.

Find the LCM of the denominators y2y^2y2, 2y2y2y, and 3y3y3y.

Rewrite each fraction with the common denominator.

Combine the numerators over the common denominator.

Simplify the numerator.

Find the LCM of the denominators 4t24t^24t2, 6t26t^26t2, and 3t3t3t.

Rewrite each fraction with the common denominator.

Combine the numerators over the common denominator.

Simplify the numerator.

Factor the denominators.

Find the LCM of the denominators (x−y)(x+y)(x-y)(x+y)(x−y)(x+y), x+yx+yx+y, and x−yx-yx−y.

Rewrite each fraction with the common denominator.

Combine the numerators over the common denominator.

Expand the terms in the numerator.

Simplify the numerator by combining like terms.

Factor the numerator and simplify the expression.

Still stuck on this one?

More Mathematics Questions

Find the least common multiple (LCM) of the denominators $5x^2$ and $2x$ .

Find the LCM of the denominators $y^2$ , $2y$ , and $3y$ .

Find the LCM of the denominators $4t^2$ , $6t^2$ , and $3t$ .

Find the LCM of the denominators $(x-y)(x+y)$ , $x+y$ , and $x-y$ .