Evaluate (7×11×19)/(3×12×15) Raised to Power -6

Q: Why does a negative exponent flip the fraction?

A negative exponent means "take the reciprocal". Since $ x^{-1} = \frac{1}{x} $, when you have $ (\frac{a}{b})^{-n} $, it becomes $ \frac{1}{(\frac{a}{b})^n} = (\frac{b}{a})^n $!

Q: Can I distribute the negative exponent to each factor?

Not correctly! While you can write $ \frac{3^6×12^6×15^6}{7^6×11^6×19^6} $, the given options show this leads to confusion. It's cleaner to flip the entire fraction first.

Q: What if I forgot to change the sign of the exponent?

You'd get $ \frac{(3×12×15)^6}{(7×11×19)^{-6}} $ which is incorrect! Remember: flipping the fraction and changing the exponent sign are both required steps.

Q: How do I know which answer choice is right?

Look for the choice that shows the original denominator on top, the original numerator on bottom, and a positive exponent 6. That's the complete transformation!

Q: Is there a shortcut for remembering this rule?

Think: "Negative exponent = flip and make positive". The fraction literally turns upside down, and the negative becomes positive. Two simple steps!

Negative Exponents with Fraction Bases

Insert the corresponding expression:

$\left(\frac{7\times11\times19}{3\times12\times15}\right)^{-6}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:13 Let's make this math problem simpler.

00:17 When a fraction is raised to a negative power, like negative N,

00:21 it's the same as the reciprocal of that fraction raised to positive N.

00:25 We'll use this idea in our exercise.

00:28 First, find the reciprocal, then use the opposite power.

00:39 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{7\times11\times19}{3\times12\times15}\right)^{-6}=$

Step-by-step solution

To solve this problem, we'll apply the rules for negative exponents:

We start with the expression: $\left(\frac{7 \times 11 \times 19}{3 \times 12 \times 15}\right)^{-6}$ .

By the rule for negative exponents, $(a/b)^{-n} = (b/a)^n$ , we invert the fraction and change the exponent from $-6$ to $6$ :

$\left(\frac{3 \times 12 \times 15}{7 \times 11 \times 19}\right)^6$

This transformation capitalizes on the idea that negative exponents reflect a reciprocal relationship.

Therefore, the expression with positive exponents is: $\left(\frac{3 \times 12 \times 15}{7 \times 11 \times 19}\right)^6$ .

Final Answer

$\left(\frac{3\times12\times15}{7\times11\times19}\right)^6$

Key Points to Remember

Essential concepts to master this topic

Rule: Negative exponent means reciprocal with positive exponent
Technique: $(a/b)^{-n} = (b/a)^n$ so flip fraction and make exponent positive
Check: Verify by ensuring final expression has positive exponent ✓

Common Mistakes

Avoid these frequent errors

Applying negative exponent to only the numerator or denominator
Don't write $\frac{7^{-6}×11^{-6}×19^{-6}}{3×12×15}$ = wrong distribution! This creates mixed positive and negative exponents. Always apply the negative exponent rule to the entire fraction as one unit.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does a negative exponent flip the fraction?

A negative exponent means "take the reciprocal". Since $x^{-1} = \frac{1}{x}$ , when you have $(\frac{a}{b})^{-n}$ , it becomes $\frac{1}{(\frac{a}{b})^n} = (\frac{b}{a})^n$ !

Can I distribute the negative exponent to each factor?

Not correctly! While you can write $\frac{3^6×12^6×15^6}{7^6×11^6×19^6}$ , the given options show this leads to confusion. It's cleaner to flip the entire fraction first.

What if I forgot to change the sign of the exponent?

You'd get $\frac{(3×12×15)^6}{(7×11×19)^{-6}}$ which is incorrect! Remember: flipping the fraction and changing the exponent sign are both required steps.

How do I know which answer choice is right?

Look for the choice that shows the original denominator on top, the original numerator on bottom, and a positive exponent 6. That's the complete transformation!

Is there a shortcut for remembering this rule?

Think: "Negative exponent = flip and make positive". The fraction literally turns upside down, and the negative becomes positive. Two simple steps!

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