Simplify the following: (3a+5b)/4 + (9+b)/8 (2x)/6 + (2x−3y)/3 + (x+y)/2 (2a+4b)/4 − 3(a−4b)/6 (3e)/4…

Question

Simplify the following: (3a+5b)/4 + (9+b)/8 (2x)/6 + (2x−3y)/3 + (x+y)/2 (2a+4b)/4 − 3(a−4b)/6 (3e)/4 + (e2)/3 (5+4)/8 − (25-2)/4

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems:

1. Simplify $\frac{3a+5b}{4} + \frac{9a+b}{8}$

Find a common denominator for 4 and 8, which is 8.
$\frac{3a+5b}{4} + \frac{9a+b}{8} = \frac{2(3a+5b)}{2 \cdot 4} + \frac{9a+b}{8}$
Rewrite the first fraction with the common denominator.
$= \frac{6a+10b}{8} + \frac{9a+b}{8}$
Add the numerators and combine like terms.
$= \frac{6a+10b+9a+b}{8} = \frac{(6a+9a)+(10b+b)}{8}$ $= \frac{15a+11b}{8}$
The simplified expression is $\boxed{\frac{15a+11b}{8}}$ .

2. Simplify $\frac{2x}{6} + \frac{2x-3y}{3} + \frac{x+y}{2}$
Find a common denominator for 6, 3, and 2, which is 6.
$\frac{2x}{6} + \frac{2x-3y}{3} + \frac{x+y}{2} = \frac{2x}{6} + \frac{2(2x-3y)}{2 \cdot 3} + \frac{3(x+y)}{3 \cdot 2}$
Rewrite each fraction with the common denominator.
$= \frac{2x}{6} + \frac{4x-6y}{6} + \frac{3x+3y}{6}$
Add the numerators and combine like terms.
$= \frac{2x+4x-6y+3x+3y}{6} = \frac{(2x+4x+3x)+(-6y+3y)}{6}$ $= \frac{9x-3y}{6}$
Factor out the common term in the numerator and simplify the fraction.
$= \frac{3(3x-y)}{6} = \frac{3x-y}{2}$
The simplified expression is $\boxed{\frac{3x-y}{2}}$ .

3. Simplify $\frac{2a+4b}{4} - \frac{3(a-4b)}{6}$
Simplify each fraction individually.
$\frac{2a+4b}{4} = \frac{2(a+2b)}{4} = \frac{a+2b}{2}$ $\frac{3(a-4b)}{6} = \frac{a-4b}{2}$
Substitute the simplified fractions back into the expression.
$\frac{a+2b}{2} - \frac{a-4b}{2}$
Subtract the numerators since they have a common denominator.
$= \frac{(a+2b)-(a-4b)}{2} = \frac{a+2b-a+4b}{2}$
Combine like terms.
$= \frac{(a-a)+(2b+4b)}{2} = \frac{6b}{2}$
Simplify the fraction.
$= 3b$
The simplified expression is $\boxed{3b}$ .

4. Simplify $\frac{3e+e^2}{4} + \frac{e}{3}$
(Assuming the operation between the two fractions is addition, as it is a common convention when an operator is missing in such problems.)
Find a common denominator for 4 and 3, which is 12.
$\frac{3e+e^2}{4} + \frac{e}{3} = \frac{3(3e+e^2)}{3 \cdot 4} + \frac{4e}{4 \cdot 3}$
Rewrite each fraction with the common denominator.
$= \frac{9e+3e^2}{12} + \frac{4e}{12}$
Add the numerators and combine like terms.
$= \frac{9e+3e^2+4e}{12} = \frac{3e^2+(9e+4e)}{12}$ $= \frac{3e^2+13e}{12}$
The simplified expression is $\boxed{\frac{3e^2+13e}{12}}$ .

5. Simplify $\frac{5 \cdot 5+4}{8} - \frac{25-2}{4}$
Calculate the numerators.
$5 \cdot 5+4 = 25+4 = 29$ $25-2 = 23$
Substitute the calculated numerators back into the expression.
$\frac{29}{8} - \frac{23}{4}$
Find a common denominator for 8 and 4, which is 8.
$= \frac{29}{8} - \frac{2 \cdot 23}{2 \cdot 4}$
Rewrite the second fraction with the common denominator.
$= \frac{29}{8} - \frac{46}{8}$
Subtract the numerators.
$= \frac{29-46}{8} = \frac{-17}{8}$
The simplified expression is $\boxed{-\frac{17}{8}}$ .

Simplify the following: (3a+5b)/4 + (9+b)/8 (2x)/6 + (2x-3y)/3 + (x+y)/2 (2a+4b)/4 - 3(a-4b)/6 (3e)/4 + (e2)/3 (5+4)/8 - (25-2)/4

Step-by-step solution

Find a common denominator for 4 and 8, which is 8.

Rewrite the first fraction with the common denominator.

Add the numerators and combine like terms.

Find a common denominator for 6, 3, and 2, which is 6.

Rewrite each fraction with the common denominator.

Add the numerators and combine like terms.

Factor out the common term in the numerator and simplify the fraction.

Simplify each fraction individually.

Substitute the simplified fractions back into the expression.

Subtract the numerators since they have a common denominator.

Combine like terms.

Simplify the fraction.

Find a common denominator for 4 and 3, which is 12.

Rewrite each fraction with the common denominator.

Add the numerators and combine like terms.

Calculate the numerators.

Substitute the calculated numerators back into the expression.

Find a common denominator for 8 and 4, which is 8.

Rewrite the second fraction with the common denominator.

Subtract the numerators.

Still stuck on this one?

More Mathematics Questions