Evaluate (2/3)^(-11): Negative Exponent Expression Solution

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

Because the exponent is applied to the entire fraction $ \frac{2}{3} $. When you raise a fraction to any power, that power affects both the top and bottom numbers equally.

Q: What's the difference between this and just flipping the fraction?

Great observation! With negative exponents, you could flip to get $ \left(\frac{3}{2}\right)^{11} $, but the question asks for the form with negative exponents in both numerator and denominator.

Q: Could I simplify this further by canceling the negative exponents?

No! The expression $ \frac{2^{-11}}{3^{-11}} $ is the correct final form. Don't try to 'cancel' the negative signs - they're part of each individual exponent.

Q: Why isn't the answer something like 11 times the reciprocal?

That would confuse multiplication with exponentiation. $ \left(\frac{2}{3}\right)^{-11} $ means the fraction multiplied by itself 11 times, not 11 times the reciprocal.

Negative Exponents with Fractional Bases

Insert the corresponding expression:

$\left(\frac{2}{3}\right)^{-11}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, a fraction raised to the power (N)

00:08 equals the numerator and denominator, each raised to the same power (N)

00:12 We will apply this formula to our exercise

00:16 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:

$\left(\frac{2}{3}\right)^{-11}=$

Final Answer

$\frac{2^{-11}}{3^{-11}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent means reciprocal: $a^{-n} = \frac{1}{a^n}$
Technique: Apply power to both numerator and denominator: $\left(\frac{2}{3}\right)^{-11} = \frac{2^{-11}}{3^{-11}}$
Check: Each part has same exponent: both 2 and 3 have exponent -11 ✓

Common Mistakes

Avoid these frequent errors

Applying negative exponent to only one part of the fraction
Don't write $\left(\frac{2}{3}\right)^{-11} = \frac{2^{-11}}{3}$ or $\frac{2}{3^{-11}}$ ! The exponent -11 applies to the entire fraction, not just one part. Always apply the exponent to both numerator AND denominator: $\frac{2^{-11}}{3^{-11}}$ .

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 apply to both the numerator and denominator?

Because the exponent is applied to the entire fraction $\frac{2}{3}$ . When you raise a fraction to any power, that power affects both the top and bottom numbers equally.

What's the difference between this and just flipping the fraction?

Great observation! With negative exponents, you could flip to get $\left(\frac{3}{2}\right)^{11}$ , but the question asks for the form with negative exponents in both numerator and denominator.

Could I simplify this further by canceling the negative exponents?

No! The expression $\frac{2^{-11}}{3^{-11}}$ is the correct final form. Don't try to 'cancel' the negative signs - they're part of each individual exponent.

Why isn't the answer something like 11 times the reciprocal?

That would confuse multiplication with exponentiation. $\left(\frac{2}{3}\right)^{-11}$ means the fraction multiplied by itself 11 times, not 11 times the reciprocal.

How can I remember which parts get the negative exponent?

Think of it this way: the exponent is like a 'hat' that sits on top of the entire fraction. Whatever is under that 'hat' gets the exponent - in this case, both 2 and 3!

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