Simplify the Fraction: -3a^(-2) ÷ -6a^(-6) Step-by-Step

Q: Why do the negative signs cancel out in the fraction?

When you have negative divided by negative, the result is positive! $ \frac{-3}{-6} = \frac{3}{6} = \frac{1}{2} $. Think of it like removing two negative signs.

Q: How do I subtract a negative exponent?

Subtracting a negative is the same as adding the positive! So $ -2 - (-6) = -2 + 6 = 4 $. The two negatives make a positive.

Q: What's the difference between multiplying and dividing with exponents?

With the same base: multiply → add exponents, divide → subtract exponents. For $ \frac{a^m}{a^n} $, you get $ a^{m-n} $.

Q: Why don't I get a negative exponent in the final answer?

Because $ -2 - (-6) = 4 $, which is positive! The negative exponent in the denominator becomes positive when you subtract it from the numerator's negative exponent.

Q: Can I rewrite this using positive exponents first?

$ a^{-2} = \frac{1}{a^2} $ and $ a^{-6} = \frac{1}{a^6} $But it's easier to use the division rule directly!

Fraction Division with Negative Exponents

Solve the following exercise:

$\frac{-3a^{-2}}{-6a^{-6}}=$

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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 A minus divided by a minus always equals a plus

00:10 When dividing powers with equal bases

00:13 The power of the result equals the difference between the powers

00:17 We'll apply this formula to our exercise, and subtract the powers

00:25 We'll divide both the numerical fraction and the unknown fraction

00:29 Let's calculate 3 divided by 6

00:43 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\frac{-3a^{-2}}{-6a^{-6}}=$

Step-by-step solution

Due to the fact that the numerator and the denominator of the fraction have terms with identical bases, we will begin by applying the law of exponents for the division of terms with identical bases:

$\frac{b^m}{b^n}=b^{m-n}$ We apply it to the problem:

$\frac{-3a^{-2}}{-6a^{-6}}=\frac{1}{2}\cdot a^{-2-(-6)}=\frac{1}{2}\cdot a^{-2+6}=\frac{1}{2}\cdot a^4$ In the first step we simplify the numerical part of the fraction. This is a simple and intuitive step as it makes it easier to work with the said fraction.

$\frac{-3a^{-2}}{-6a^{-6}}=\frac{-3}{-6}\cdot\frac{a^{-2}}{a^{-6}}=\frac{1}{2}\cdot\frac{a^{-2}}{a^{-6}}=\ldots$ We then return to the problem and subsequently obtain the following expression:

$\frac{1}{2}\cdot a^4$ Therefore, the correct answer is option C.

Final Answer

$\frac{1}{2}a^4$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract exponents
Technique: $a^{-2} ÷ a^{-6} = a^{-2-(-6)} = a^4$
Check: Substitute values to verify $\frac{1}{2}a^4$ matches original expression ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting when dividing
Don't add exponents like -2 + (-6) = -8 when dividing = wrong power! Division requires subtracting the denominator exponent from the numerator exponent. Always subtract: -2 - (-6) = -2 + 6 = 4.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do the negative signs cancel out in the fraction?

When you have negative divided by negative, the result is positive! $\frac{-3}{-6} = \frac{3}{6} = \frac{1}{2}$ . Think of it like removing two negative signs.

How do I subtract a negative exponent?

Subtracting a negative is the same as adding the positive! So $-2 - (-6) = -2 + 6 = 4$ . The two negatives make a positive.

What's the difference between multiplying and dividing with exponents?

With the same base: multiply → add exponents, divide → subtract exponents. For $\frac{a^m}{a^n}$ , you get $a^{m-n}$ .

Why don't I get a negative exponent in the final answer?

Because $-2 - (-6) = 4$ , which is positive! The negative exponent in the denominator becomes positive when you subtract it from the numerator's negative exponent.

Can I rewrite this using positive exponents first?

$a^{-2} = \frac{1}{a^2}$ and $a^{-6} = \frac{1}{a^6}$
But it's easier to use the division rule directly!

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