Combine like terms containing fractional coefficients and rational exponents: $\frac{3}{4}x^{\frac{2}{3}}y - \frac{2}{5}xy^{\frac{1}{2}} + \frac{5}{6}x^{\frac{2}{3}}y - \frac{7}{10}xy^{\frac{1}{2}} + \frac{1}{3}x^{\frac{2}{3}}y$. Express the final simplified result with irreducible fractional coefficients.

Combine like terms containing fractional coefficients and rational exponents: $\frac{3}{4}x^{\frac{2}{3}}y - \frac{2}{5}xy^{\frac{1}{2}} + \frac{5}{6}x^{\frac{2}{3}}y - \frac{7}{10}xy^{\frac{1}{2}} + \frac{1}{3}x^{\frac{2}{3}}y$ . Express the final simplified result with irreducible fractional coefficients.

Final answer

$\frac{23}{12}x^{\frac{2}{3}}y - \frac{11}{10}xy^{\frac{1}{2}}$

Steps

Identify and group like terms

To simplify the algebraic expression, we group like terms together and combine their numerical coefficients.

Like terms must have identical variables raised to identical exponents. In the given expression: $\frac{3}{4}x^{\frac{2}{3}}y - \frac{2}{5}xy^{\frac{1}{2}} + \frac{5}{6}x^{\frac{2}{3}}y - \frac{7}{10}xy^{\frac{1}{2}} + \frac{1}{3}x^{\frac{2}{3}}y$

There are two groups of like terms:

Terms containing $x^{\frac{2}{3}}y$ : $\left(\frac{3}{4} + \frac{5}{6} + \frac{1}{3}\right)x^{\frac{2}{3}}y$
Terms containing $xy^{\frac{1}{2}}$ : $\left(-\frac{2}{5} - \frac{7}{10}\right)xy^{\frac{1}{2}}$

Combine the coefficients for each group

For the $x^{\frac{2}{3}}y$ terms: Find the least common denominator ( $\text{LCD}$ ) for the denominators $4$ , $6$ , and $3$ , which is $12$ : $\frac{3}{4} = \frac{3 \cdot 3}{4 \cdot 3} = \frac{9}{12}$ $\frac{5}{6} = \frac{5 \cdot 2}{6 \cdot 2} = \frac{10}{12}$ $\frac{1}{3} = \frac{1 \cdot 4}{3 \cdot 4} = \frac{4}{12}$

Add the numerators: $\frac{9}{12} + \frac{10}{12} + \frac{4}{12} = \frac{9 + 10 + 4}{12} = \frac{23}{12}$

This gives $\frac{23}{12}x^{\frac{2}{3}}y$ .
For the $xy^{\frac{1}{2}}$ terms: Find the least common denominator for $5$ and $10$ , which is $10$ : $-\frac{2}{5} = -\frac{2 \cdot 2}{5 \cdot 2} = -\frac{4}{10}$

Combine the numerators: $-\frac{4}{10} - \frac{7}{10} = \frac{-4 - 7}{10} = -\frac{11}{10}$

This gives $-\frac{11}{10}xy^{\frac{1}{2}}$ .

Write the final simplified expression

Combining the two parts yields: $\frac{23}{12}x^{\frac{2}{3}}y - \frac{11}{10}xy^{\frac{1}{2}}$

Common Pitfall: When combining like terms, only add or subtract the numerical coefficients. Do not alter or add the exponents of the variables.

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student adds the numerators and denominators across fractions instead of finding a common denominator when combining coefficients.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\left(\frac{3}{4} + \frac{5}{6} + \frac{1}{3}\right)$ x^{ $\frac{2}{3}$ }y + $\left(-\frac{2}{5} - \frac{7}{10}\right)$ xy^{ $\frac{1}{2}$ }
Step 2
$\frac{3+5+1}{4+6+3}x^{\frac{2}{3}}y + \left(\frac{-2-7}{5+10}\right)xy^{\frac{1}{2}}$
Correct
$\left(\frac{9}{12} + \frac{10}{12} + \frac{4}{12}\right)$ x^{ $\frac{2}{3}$ }y + $\left(-\frac{4}{10} - \frac{7}{10}\right)$ xy^{ $\frac{1}{2}$ }
To add or subtract fractions, you must rewrite them with a common denominator first. Adding numerators together and denominators together across different fractions is not mathematically valid.
Step 3
$\frac{9}{13}x^{\frac{2}{3}}y - \frac{9}{15}xy^{\frac{1}{2}} \implies \frac{9}{13}x^{\frac{2}{3}}y - \frac{3}{5}xy^{\frac{1}{2}}$
Correct
$\frac{23}{12}x^{\frac{2}{3}}y - \frac{11}{10}xy^{\frac{1}{2}}$
Although you simplified $\frac{9}{15}$ to $\frac{3}{5}$ correctly, the final coefficients are incorrect because they were calculated from an invalid fraction addition rule.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly identified the like terms with matching fractional exponents and grouped their coefficients together.

You simplified the resulting fraction $-\frac{9}{15}$ to lowest terms as $-\frac{3}{5}$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Fraction addition without common denominator

none

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated fraction addition as adding numerators across numerators and denominators across denominators ( $\frac{a}{b} + \frac{c}{d} = \frac{a+c}{b+d}$ ), rather than converting each fraction to an equivalent form over a least common denominator.

REASONING PATH

Your ACEpath teacher’s suggested approach

Find the Least Common Denominator (LCD): Fractions can only be combined by addition or subtraction when their denominators are identical: $\frac{a}{d} + \frac{b}{d} = \frac{a+b}{d}$ .
Find the least common multiple of all denominators in the group before combining numerators.
Convert to Equivalent Fractions: Multiply the numerator and denominator by the same non-zero integer so that the denominator equals the LCD: $\frac{a}{b} = \frac{a \cdot k}{b \cdot k}$ .
Rewrite each fraction over the LCD, then add or subtract only the numerators while keeping the denominator unchanged.

CORRECTED PATH

ACEpath Teacher's Demonstration

For $x^{\frac{2}{3}}y$ : The LCD of $4$ , $6$ , and $3$ is $12$ . Convert each fraction: $\frac{3}{4} = \frac{9}{12}$ , $\frac{5}{6} = \frac{10}{12}$ , and $\frac{1}{3} = \frac{4}{12}$ . Add the numerators: $\frac{9 + 10 + 4}{12} = \frac{23}{12}$ .
For $xy^{\frac{1}{2}}$ : The LCD of $5$ and $10$ is $10$ . Convert $-\frac{2}{5} = -\frac{4}{10}$ . Combine the numerators: $\frac{-4 - 7}{10} = -\frac{11}{10}$ .

Final answer

$\frac{23}{12}x^{\frac{2}{3}}y - \frac{11}{10}xy^{\frac{1}{2}}$

NEXT STEPS

Final Takeaways from Your Teacher

Review adding fractions with unequal denominators by always writing out the LCD first.
Practice combining like terms where coefficients are fractions to keep algebra and fraction arithmetic clearly separated.

OverviewACEpath Teacher's Diagnosis

02 The student mishandles the subtraction when combining negative coefficients, adding the absolute values but writing a positive result or subtracting -4 - 7 incorrectly as -3/10.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
( $\frac{3}{4}$ + $\frac{5}{6}$ + $\frac{1}{3}$ )x^{ $\frac{2}{3}$ }y + (- $\frac{2}{5}$ - $\frac{7}{10}$ )xy^{ $\frac{1}{2}$ }
Step 2
= ( $\frac{9}{12}$ + $\frac{10}{12}$ + $\frac{4}{12}$ )x^{ $\frac{2}{3}$ }y + (- $\frac{4}{10}$ - $\frac{7}{10}$ )xy^{ $\frac{1}{2}$ }
Step 3
= $\frac{23}{12}$ x^{ $\frac{2}{3}$ }y - $\frac{3}{10}$ xy^{ $\frac{1}{2}$ }
Correct
$\frac{23}{12}x^{\frac{2}{3}}y - \frac{11}{10}xy^{\frac{1}{2}}$
You combined $9 + 10 + 4 = 23$ correctly, but evaluated $-\frac{4}{10} - \frac{7}{10}$ as $-\frac{3}{10}$ . Subtracting $7$ from $-4$ yields $-4 - 7 = -11$ , giving $-\frac{11}{10}xy^{\frac{1}{2}}$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly identified which terms are like terms based on identical variable powers.

You found the correct least common denominator for both groups of rational coefficients.

You correctly added the positive numerators to obtain $\frac{23}{12}x^{\frac{2}{3}}y$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Signed Number Subtraction Error

You calculated $-4 - 7$ as $-3$ (taking the difference $7 - 4$ ) rather than accumulating the two negative values.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the addition of two negative numbers with the subtraction of their absolute values: when combining $-a - b$ , the magnitudes add together to give $-(a + b)$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Combine Same-Sign Integers: $-a - b = -(a + b)$
Add the absolute values and keep the common negative sign: $-4 - 7 = -(4 + 7) = -11$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Combine the numerators over the common denominator $10$ : $\frac{-4 - 7}{10} = -\frac{11}{10}$ .
Attach the variable part: $-\frac{11}{10}xy^{\frac{1}{2}}$ .

Final answer

$\frac{23}{12}x^{\frac{2}{3}}y - \frac{11}{10}xy^{\frac{1}{2}}$

NEXT STEPS

Final Takeaways from Your Teacher

When both numerators are negative, rewrite them explicitly as $-(4 + 7)$ to prevent accidental subtraction of absolute values.
Double-check that moving further left on the number line always decreases the value.

OverviewACEpath Teacher's Diagnosis

03 The student erroneously treats all terms as like terms and combines all five coefficients into a single term while adding their variable exponents.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\left(\frac{3}{4} - \frac{2}{5} + \frac{5}{6} - \frac{7}{10} + \frac{1}{3}\right)$ x^{ $\frac{2}{3}$ +1+ $\frac{2}{3}$ +1+ $\frac{2}{3}$ } y^{1+ $\frac{1}{2}$ +1+ $\frac{1}{2}$ +1}
Correct
$\left(\frac{3}{4} + \frac{5}{6} + \frac{1}{3}\right)$ x^{ $\frac{2}{3}$ }y + $\left(-\frac{2}{5} - \frac{7}{10}\right)$ xy^{ $\frac{1}{2}$ }
You combined all five terms into a single term and added their variable exponents together. In addition, terms can only be combined if they have the exact same variable part (same bases and same exponents), and their exponents do not change.
Step 2
$\left(\frac{45}{60} - \frac{24}{60} + \frac{50}{60} - \frac{42}{60} + \frac{20}{60}\right) x^{\frac{11}{3}} y^4 \implies \frac{49}{60} x^{\frac{11}{3}} y^4$
Correct
$\frac{23}{12}x^{\frac{2}{3}}y - \frac{11}{10}xy^{\frac{1}{2}}$
Although your arithmetic using the common denominator $60$ was executed well, this step is affected by the initial error of merging unlike terms and altering the exponents.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly identified a common denominator of $60$ to combine fractions with denominators $4$ , $5$ , $6$ , $10$ , and $3$ .

You accurately converted the fractional coefficients to equivalent fractions with a shared denominator.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Combining unlike terms and adding exponents during addition

You grouped all five coefficients into one single term and added exponents across different variables.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You applied the product rule of exponents ( $x^a \cdot x^b = x^{a+b}$ ) to an addition problem. Terms being added can only be combined if their variable parts are identical, and combining like terms only alters their numerical coefficients while leaving exponents unchanged.

REASONING PATH

Your ACEpath teacher’s suggested approach

Identify Like Terms: Two terms are like terms if and only if they share identical variables raised to identical powers.
Separate the expression into two distinct groups: terms containing $x^{\frac{2}{3}}y$ and terms containing $xy^{\frac{1}{2}}$ .
Keep Exponents Constant: When adding or subtracting like terms, add only the coefficients: $a u + b u = (a+b)u$ .
Never add or change exponents when performing addition or subtraction.

CORRECTED PATH

ACEpath Teacher's Demonstration

Group like terms with identical variable factors: $\left(\frac{3}{4} + \frac{5}{6} + \frac{1}{3}\right)x^{\frac{2}{3}}y + \left(-\frac{2}{5} - \frac{7}{10}\right)xy^{\frac{1}{2}}$ .
Find common denominators for each group separately: $\text{LCD}(4, 6, 3) = 12$ and $\text{LCD}(5, 10) = 10$ .
Combine coefficients within each group: $\left(\frac{9}{12} + \frac{10}{12} + \frac{4}{12}\right)x^{\frac{2}{3}}y = \frac{23}{12}x^{\frac{2}{3}}y$ and $\left(-\frac{4}{10} - \frac{7}{10}\right)xy^{\frac{1}{2}} = -\frac{11}{10}xy^{\frac{1}{2}}$ .

Final answer

$\frac{23}{12}x^{\frac{2}{3}}y - \frac{11}{10}xy^{\frac{1}{2}}$

NEXT STEPS

Final Takeaways from Your Teacher

Separate distinct variable structures before performing any fraction arithmetic.
Keep exponents untouched when adding or subtracting algebraic terms; modify only their coefficients.

Lock it in

Question

What is the core rule for adding two fractions $\frac{a}{b} + \frac{c}{d}$ with different denominators?

Answer

Find a common denominator first: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ . Never add the denominators together.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student adds the numerators and denominators across fractions instead of finding a common denominator when combining coefficients.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student mishandles the subtraction when combining negative coefficients, adding the absolute values but writing a positive result or subtracting -4 - 7 incorrectly as -3/10.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student erroneously treats all terms as like terms and combines all five coefficients into a single term while adding their variable exponents.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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