Solve: mn-(3mn-n/m:m/n+nm/3:m²) Complex Algebraic Expression

Q: What does the colon symbol (:) mean in algebra?

The colon (:) represents division! So $ a : b $ means $ a \div b $ or $ \frac{a}{b} $. It's the same as the division symbol ÷.

Q: How do I divide fractions like n/m : m/n?

To divide fractions, multiply by the reciprocal! So $ \frac{n}{m} : \frac{m}{n} = \frac{n}{m} \times \frac{n}{m} = \frac{n^2}{m^2} $. Remember: flip the second fraction and multiply.

Q: Why do we distribute the negative sign outside the parentheses?

The negative sign in front of parentheses means multiply everything inside by -1. So $ -(3mn - \frac{n^2}{m^2} + \frac{n}{3m}) $ becomes $ -3mn + \frac{n^2}{m^2} - \frac{n}{3m} $.

Q: How do I combine like terms with mn?

Look for terms with the same variables and exponents. Here, $ mn - 3mn = -2mn $ because they both have exactly mn (no other like terms exist in this problem).

Q: Can I simplify this expression further?

No, $ -2mn + \frac{n^2}{m^2} - \frac{n}{3m} $ is fully simplified! Each term has different combinations of variables, so they cannot be combined further.

Q: What if I get confused with the order of operations?

Follow PEMDAS: Parentheses first, then handle divisions (like :), then addition/subtraction from left to right. Work step-by-step and don't rush!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:09 Division is also multiplication by the reciprocal

00:20 Move the division to the denominator

00:28 Make sure to multiply numerator by numerator and denominator by denominator

00:31 Simplify what's possible

00:40 Negative times positive is always negative

00:44 Negative times negative is always positive

00:50 Negative times positive is always negative

00:57 Collect like terms

01:05 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

$mn-(3mn-\frac{n}{m}:\frac{m}{n}+\frac{nm}{3}:m^2)=\text{?}$

2

Step-by-step solution

To solve this expression, we will simplify it step by step:

The expression to simplify is:

mn - \left(3mn - \frac{n}{m} : \frac{m}{n} + \frac{nm}{3} : m^2 \right)

First, simplify each term within the parentheses:

The division $\frac{n}{m} : \frac{m}{n}$ can be rewritten as $\frac{n}{m} \cdot \frac{n}{m}$ because $\frac{m}{n}$ inverted becomes $\frac{n}{m}$ . This results in $\frac{n^2}{m^2}$ .
Next, the operation $\frac{nm}{3} : m^2$ is rewritten as $\frac{nm}{3} \cdot \frac{1}{m^2}$ . This simplifies to $\frac{n}{3m}$ because $m \cdot m = m^2$ .

Substitute these simplified forms back into the expression inside the parentheses:

3mn - \frac{n^2}{m^2} + \frac{n}{3m}

Now rewrite the entire expression outside of the parentheses and solve:

mn - \left( 3mn - \frac{n^2}{m^2} + \frac{n}{3m} \right)

Use distributive law to expand:

mn - 3mn + \frac{n^2}{m^2} - \frac{n}{3m}

Combine like terms:

-2mn + \frac{n^2}{m^2} - \frac{n}{3m}

Therefore, the solution to the simplified expression is:

$-2mn + \frac{n^2}{m^2} - \frac{n}{3m}$

3

Final Answer

$-2mn+\frac{n^2}{m^2}-\frac{n}{3m}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: Convert a:b to a×(1/b) or a÷b
Technique: $\frac{n}{m} : \frac{m}{n} = \frac{n}{m} \times \frac{n}{m} = \frac{n^2}{m^2}$
Check: Substitute values to verify $-2mn + \frac{n^2}{m^2} - \frac{n}{3m}$ ✓

Common Mistakes

Avoid these frequent errors

Confusing division notation with multiplication
Don't treat a:b as a×b = wrong operations! The colon (:) means division, not multiplication, leading to completely incorrect simplifications. Always convert a:b to a÷b or a×(1/b) first.

Practice Quiz

Test your knowledge with interactive questions

$70:(14\times5)=$

FAQ

Everything you need to know about this question

What does the colon symbol (:) mean in algebra?

+

The colon (:) represents division! So $a : b$ means $a \div b$ or $\frac{a}{b}$ . It's the same as the division symbol ÷.

How do I divide fractions like n/m : m/n?

+

To divide fractions, multiply by the reciprocal! So $\frac{n}{m} : \frac{m}{n} = \frac{n}{m} \times \frac{n}{m} = \frac{n^2}{m^2}$ . Remember: flip the second fraction and multiply.

Why do we distribute the negative sign outside the parentheses?

+

The negative sign in front of parentheses means multiply everything inside by -1. So $-(3mn - \frac{n^2}{m^2} + \frac{n}{3m})$ becomes $-3mn + \frac{n^2}{m^2} - \frac{n}{3m}$ .

How do I combine like terms with mn?

+

Look for terms with the same variables and exponents. Here, $mn - 3mn = -2mn$ because they both have exactly mn (no other like terms exist in this problem).

Can I simplify this expression further?

+

No, $-2mn + \frac{n^2}{m^2} - \frac{n}{3m}$ is fully simplified! Each term has different combinations of variables, so they cannot be combined further.

What if I get confused with the order of operations?

+

Follow PEMDAS: Parentheses first, then handle divisions (like :), then addition/subtraction from left to right. Work step-by-step and don't rush!

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Solve: mn-(3mn-n/m:m/n+nm/3:m²) Complex Algebraic Expression

Complex Algebraic Expressions with Division Operations

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