Verify the Equation: (a²b)/(ac) = (ab)/c in Algebraic Fractions

Q: Why can I cancel the 'a' terms in this fraction?

You can cancel because division by the same non-zero value doesn't change the fraction's value! Since $ a^2 = a \cdot a $, we have one $ a $ in both numerator and denominator to cancel.

Q: What if a = 0? Does this equation still work?

If $ a = 0 $, both sides become $ \frac{0}{c} = 0 $ (assuming $ c \neq 0 $), so the equation is still true! However, we must assume $ a \neq 0 $ and $ c \neq 0 $ for proper cancellation.

Q: Is this the same as reducing fractions with numbers?

Yes, exactly! Just like $ \frac{6}{9} = \frac{2 \cdot 3}{3 \cdot 3} = \frac{2}{3} $, we cancel common factors. Here we cancel the common factor $ a $ from numerator and denominator.

Q: Can I use this method with more complex expressions?

Absolutely! This factor-and-cancel method works with any algebraic fractions. Look for common factors in numerator and denominator, then cancel them out systematically.

Algebraic Fraction Simplification with Exponents

Indicate whether true or false

$\frac{a^2\cdot b}{a\cdot c}=\frac{a\cdot b}{c}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine if the equation is correct

00:03 Break down the exponent into products

00:08 Simplify what we can

00:14 Now let's compare the expressions

00:23 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Indicate whether true or false

$\frac{a^2\cdot b}{a\cdot c}=\frac{a\cdot b}{c}$

Step-by-step solution

Let's examine the problem first:

$\frac{a^2\cdot b}{a\cdot c} \stackrel{?}{= }\frac{a\cdot b}{c}$ Note that we can simplify the expression on the left side, this can be done by reducing the fraction, for this, let's recall the definition of exponents:

$\frac{\textcolor{red}{a^2}\cdot b}{a\cdot c} =\\ \frac{\textcolor{red}{\not{a}}\cdot a\cdot b}{\not{a}\cdot c}=\\ \boxed{\frac{ a\cdot b}{c}}\\$ The expression on the right side is also:

$\frac{a\cdot b}{c}$ Therefore the expressions on both sides of the equation (assumed to be true) are indeed equal, meaning:

$\frac{a^2\cdot b}{a\cdot c}= \frac{a\cdot b}{c} \stackrel{!}{= }\frac{a\cdot b}{c}$

(In other words, an identity equation holds- which is true for all possible values of the parameters $a,b,c$ )

Therefore, the correct answer is answer A.

Final Answer

True

Key Points to Remember

Essential concepts to master this topic

Rule: Cancel common factors by dividing numerator and denominator
Technique: Rewrite $a^2 = a \cdot a$ to see cancellation clearly
Check: Both sides equal $\frac{ab}{c}$ when simplified ✓

Common Mistakes

Avoid these frequent errors

Incorrectly canceling exponents
Don't subtract exponents like $\frac{a^2}{a} = a^{2-1}$ without understanding! This mechanical approach misses the actual cancellation happening. Always see $a^2$ as $a \cdot a$ first, then cancel one $a$ from top and bottom.

Practice Quiz

Test your knowledge with interactive questions

Determine if the simplification shown below is correct:

$\frac{7}{7\cdot8}=8$

FAQ

Everything you need to know about this question

Why can I cancel the 'a' terms in this fraction?

You can cancel because division by the same non-zero value doesn't change the fraction's value! Since $a^2 = a \cdot a$ , we have one $a$ in both numerator and denominator to cancel.

What if a = 0? Does this equation still work?

If $a = 0$ , both sides become $\frac{0}{c} = 0$ (assuming $c \neq 0$ ), so the equation is still true! However, we must assume $a \neq 0$ and $c \neq 0$ for proper cancellation.

Is this the same as reducing fractions with numbers?

Yes, exactly! Just like $\frac{6}{9} = \frac{2 \cdot 3}{3 \cdot 3} = \frac{2}{3}$ , we cancel common factors. Here we cancel the common factor $a$ from numerator and denominator.

How do I know when fractions are equal without calculating?

Two fractions are equal if one can be simplified to match the other! Always simplify the more complex fraction first by canceling common factors, then compare the results.

Can I use this method with more complex expressions?

Absolutely! This factor-and-cancel method works with any algebraic fractions. Look for common factors in numerator and denominator, then cancel them out systematically.

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