Solve 3-(12a-(a:14/a+12)): Complex Algebraic Expression Simplification

Q: Why does dividing by a fraction mean multiplying by its reciprocal?

Think of it this way: dividing by $ \frac{1}{2} $ means 'how many halves fit into something?' The answer is twice as many as whole units! So $ a ÷ \frac{14}{a} = a \times \frac{a}{14} $.

Q: How do I handle the nested parentheses correctly?

Work from the innermost parentheses outward. First solve $ a : \frac{14}{a} + 12 $, then subtract from 12a, finally subtract the whole thing from 3.

Q: When do I distribute the negative sign?

Distribute the negative when you have subtraction in front of parentheses. In $ 3 - (12a - \frac{a^2}{14} - 12) $, the negative flips all signs inside!

Q: Why is my final answer different from the choices?

Double-check your order of operations and make sure you converted the division correctly. The trickiest part is $ a : \frac{14}{a} = \frac{a^2}{14} $, not $ \frac{14}{a^2} $.

Q: Can I solve this by substituting a number for 'a'?

Yes, for checking! Try a = 2: the original gives $ 3 - (24 - (\frac{4}{14} + 12)) = 15 - 24 + \frac{4}{14} $. This should match your algebraic answer.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Negative times positive is always negative

00:10 Negative times negative is always positive

00:23 Collect like terms

00:31 Division is also multiplication by the reciprocal

00:46 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

$3-(12a-(a:\frac{14}{a}+12))=\text{?}$

2

Step-by-step solution

To solve this problem, follow these steps:

Start by simplifying the innermost operations and move outward:

Step 1: Simplify the division inside the parentheses:
$a : \frac{14}{a} = \frac{a \cdot a}{14} = \frac{a^2}{14}$
Step 2: Substitute this back into the expression:
$12a - \left( \frac{a^2}{14} + 12 \right)$
Step 3: Simplify inside the parenthesis:
$12a - \left( \frac{a^2}{14} + 12 \right) = 12a - \frac{a^2}{14} - 12$
Step 4: Distribute the negative sign across the expression within parentheses:
$3 - (12a - \frac{a^2}{14} - 12) = 3 - 12a + \frac{a^2}{14} + 12$
Step 5: Combine the constant terms:
$3 + 12 - 12a + \frac{a^2}{14} = 15 - 12a + \frac{a^2}{14}$

Thus, the expression simplifies to $15 - 12a + \frac{a^2}{14}$ .

3

Final Answer

$15-12a+\frac{a^2}{14}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: Division by fraction equals multiplication by reciprocal
Technique: $a : \frac{14}{a} = a \times \frac{a}{14} = \frac{a^2}{14}$
Check: Work from inside out, distribute negative signs carefully ✓

Common Mistakes

Avoid these frequent errors

Treating division by fraction incorrectly
Don't compute $a : \frac{14}{a}$ as $\frac{a}{14}$ = wrong result! This ignores the division rule completely. Always convert division by fraction to multiplication by reciprocal first.

Practice Quiz

Test your knowledge with interactive questions

$$ 100-(5+55)= $$

FAQ

Everything you need to know about this question

Why does dividing by a fraction mean multiplying by its reciprocal?

+

Think of it this way: dividing by $\frac{1}{2}$ means 'how many halves fit into something?' The answer is twice as many as whole units! So $a ÷ \frac{14}{a} = a \times \frac{a}{14}$ .

How do I handle the nested parentheses correctly?

+

Work from the innermost parentheses outward. First solve $a : \frac{14}{a} + 12$ , then subtract from 12a, finally subtract the whole thing from 3.

When do I distribute the negative sign?

+

Distribute the negative when you have subtraction in front of parentheses. In $3 - (12a - \frac{a^2}{14} - 12)$ , the negative flips all signs inside!

Why is my final answer different from the choices?

+

Double-check your order of operations and make sure you converted the division correctly. The trickiest part is $a : \frac{14}{a} = \frac{a^2}{14}$ , not $\frac{14}{a^2}$ .

Can I solve this by substituting a number for 'a'?

+

Yes, for checking! Try a = 2: the original gives $3 - (24 - (\frac{4}{14} + 12)) = 15 - 24 + \frac{4}{14}$ . This should match your algebraic answer.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Commutative, Distributive and Associative Properties questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Solve 3-(12a-(a:14/a+12)): Complex Algebraic Expression Simplification

Order of Operations with Division by Fractions

❤️ Continue Your Math Journey!