Evaluate (5/594)^(-3): Negative Exponent Expression Solution

Q: Why are all three answer choices correct?

Each form represents a different stage of simplification of the same expression. They're all mathematically equivalent - just like how 2+2, 4, and 8/2 all equal the same value!

Q: Which form should I use in my final answer?

Any of the three forms is acceptable! However, $ \left(\frac{6 \times 9 \times 11}{5}\right)^3 $ is often preferred because it's the most direct result of applying the negative exponent rule.

Q: How do I remember the negative exponent rule?

Think "flip and drop the negative" - when you see a negative exponent, flip the fraction (numerator becomes denominator and vice versa) and make the exponent positive.

Q: Can I leave the answer as separate factors like 6³×9³×11³?

Yes! $ \frac{6^3 \times 9^3 \times 11^3}{5^3} $ is a completely valid form. Sometimes expanding the factors makes it easier to see patterns or simplify further.

Negative Exponents with Multiple Equivalent Forms

Insert the corresponding expression:

$\left(\frac{5}{6\times9\times11}\right)^{-3}=$

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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 a negative exponent (-N)

00:07 is equal to its reciprocal raised to the opposite exponent (N)

00:12 We will apply this formula to our exercise

00:15 We'll convert to the reciprocal number and raise it to the opposite power

00:24 According to the laws of exponents, a fraction raised to an exponent (N)

00:28 is equal to the fraction where both the numerator and denominator are raised to the power (N)

00:33 We will apply this formula to our exercise

00:37 We'll raise both the numerator and denominator to the appropriate power, maintaining the parentheses

00:41 According to the laws of exponents, a product raised to the exponent (N)

00:44 is equal to the product broken down into factors where each factor is raised to the power (N)

00:49 We will apply this formula to our exercise

00:52 We'll break down each product into factors and raise them to the appropriate power

00:57 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:

$\left(\frac{5}{6\times9\times11}\right)^{-3}=$

Step-by-step solution

To solve this problem, we will employ exponent rules:

Convert the negative exponent: $\left(\frac{5}{6 \times 9 \times 11}\right)^{-3} = \left(\frac{6 \times 9 \times 11}{5}\right)^3$ .
Simplify using the power of a quotient: $\left(\frac{6 \times 9 \times 11}{5}\right)^3 = \frac{(6 \times 9 \times 11)^3}{5^3}$ .
Further expand separately: $\frac{6^3 \times 9^3 \times 11^3}{5^3}$ .

Therefore, each proposed expression $\frac{6^3 \times 9^3 \times 11^3}{5^3}$ , $\frac{(6 \times 9 \times 11)^3}{5^3}$ , and $\left(\frac{6 \times 9 \times 11}{5}\right)^3$ are equivalent and correct interpretations of the original expression.

All answers are correct.

The correct choice is option 4: All answers are correct.

Final Answer

All answers are correct

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent flips fraction and makes exponent positive
Technique: $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ transforms the base
Check: All three forms should give identical numerical values ✓

Common Mistakes

Avoid these frequent errors

Applying negative exponent to only part of the expression
Don't just change $\left(\frac{5}{594}\right)^{-3}$ to $\frac{5^{-3}}{594}$ = wrong application! The negative exponent applies to the entire fraction, not individual parts. Always flip the entire fraction first, then apply the positive exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are all three answer choices correct?

Each form represents a different stage of simplification of the same expression. They're all mathematically equivalent - just like how 2+2, 4, and 8/2 all equal the same value!

Which form should I use in my final answer?

Any of the three forms is acceptable! However, $\left(\frac{6 \times 9 \times 11}{5}\right)^3$ is often preferred because it's the most direct result of applying the negative exponent rule.

How do I remember the negative exponent rule?

Think "flip and drop the negative" - when you see a negative exponent, flip the fraction (numerator becomes denominator and vice versa) and make the exponent positive.

Can I leave the answer as separate factors like 6³×9³×11³?

Yes! $\frac{6^3 \times 9^3 \times 11^3}{5^3}$ is a completely valid form. Sometimes expanding the factors makes it easier to see patterns or simplify further.

What if I calculated 594 = 6×9×11 wrong?

Double-check your multiplication: 6 × 9 = 54, then 54 × 11 = 594. Getting this factorization correct is crucial for recognizing why all answer forms are equivalent.

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