Solve ((4×5)/(3×2))^(-2): Negative Exponent with Complex Fractions

Q: Why does the negative exponent go to both the numerator and denominator?

Because the negative exponent applies to the entire fraction $ \left(\frac{4\times5}{3\times2}\right)^{-2} $. Think of it as $ (\text{whole fraction})^{-2} $, so every part gets the exponent.

Q: What's the difference between the correct and incorrect answers?

The correct answer has $ 4^{-2}\times5^{-2} $ in the numerator AND $ 3^{-2}\times2^{-2} $ in the denominator. Wrong answers only apply the negative exponent to some factors, not all.

Q: Can I simplify this further?

Yes! You could calculate $ \frac{4\times5}{3\times2} = \frac{20}{6} = \frac{10}{3} $ first, then apply $ \left(\frac{10}{3}\right)^{-2} = \left(\frac{3}{10}\right)^{2} $. But the question asks for the expression, not the final number.

Q: How do I remember which way the fraction flips?

Think: negative exponent = flip and make positive. So $ \left(\frac{\text{top}}{\text{bottom}}\right)^{-n} $ becomes $ \left(\frac{\text{bottom}}{\text{top}}\right)^{n} $, then distribute the exponent!

Q: Why can't I just make some exponents positive and others negative?

Because that would mean you're applying different rules to the same exponent! The -2 applies equally to every factor. Mixing positive and negative exponents would create an entirely different expression.

Negative Exponents with Fraction Bases

Insert the corresponding expression:

$\left(\frac{4\times5}{3\times2}\right)^{-2}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, a fraction (A\B) raised to a power (N)

00:07 is equal to the numerator and denominator raised to the same power (N)

00:12 We will apply this formula to our exercise

00:15 We'll raise both the numerator and denominator to the power (N)

00:27 According to the laws of exponents when a product is raised to a power (N)

00:32 it is equal to each factor in the product separately raised to the same power (N)

00:40 We will apply this formula to our exercise

00:50 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{4\times5}{3\times2}\right)^{-2}=$

Final Answer

$\frac{4^{-2}\times5^{-2}}{3^{-2}\times2^{-2}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent makes the entire fraction flip and become positive
Technique: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$ then distribute exponent
Check: Each factor gets the same exponent: $\frac{4^{-2}\times5^{-2}}{3^{-2}\times2^{-2}}$ ✓

Common Mistakes

Avoid these frequent errors

Only applying negative exponent to numerator or denominator
Don't just change 4×5 to 4^{-2}×5^{-2} and leave 3×2 unchanged = wrong distribution! The negative exponent applies to the entire fraction, so both numerator AND denominator get the negative exponent. Always distribute the exponent to every factor in both parts.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does the negative exponent go to both the numerator and denominator?

Because the negative exponent applies to the entire fraction $\left(\frac{4\times5}{3\times2}\right)^{-2}$ . Think of it as $(\text{whole fraction})^{-2}$ , so every part gets the exponent.

What's the difference between the correct and incorrect answers?

The correct answer has $4^{-2}\times5^{-2}$ in the numerator AND $3^{-2}\times2^{-2}$ in the denominator. Wrong answers only apply the negative exponent to some factors, not all.

Can I simplify this further?

Yes! You could calculate $\frac{4\times5}{3\times2} = \frac{20}{6} = \frac{10}{3}$ first, then apply $\left(\frac{10}{3}\right)^{-2} = \left(\frac{3}{10}\right)^{2}$ . But the question asks for the expression, not the final number.

How do I remember which way the fraction flips?

Think: negative exponent = flip and make positive. So $\left(\frac{\text{top}}{\text{bottom}}\right)^{-n}$ becomes $\left(\frac{\text{bottom}}{\text{top}}\right)^{n}$ , then distribute the exponent!

Why can't I just make some exponents positive and others negative?

Because that would mean you're applying different rules to the same exponent! The -2 applies equally to every factor. Mixing positive and negative exponents would create an entirely different expression.

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