Simplify the Expression: a⁴/a⁻⁶ Using Exponent Rules

Q: What happens to the negative exponent in the denominator?

The negative exponent stays with the base when you apply the division rule. You're subtracting the entire exponent $ -6 $, not just the 6.

Q: Can I move the negative exponent to the numerator instead?

Yes! $ \frac{a^4}{a^{-6}} = a^4 \times a^6 = a^{10} $. Moving $ a^{-6} $ up makes it $ a^6 $, then multiply the powers.

Q: How do I remember the division rule?

Think "top minus bottom" for division: $ \frac{a^m}{a^n} = a^{m-n} $. Always subtract the denominator exponent from the numerator exponent.

Q: What if both exponents were negative?

Same rule applies! For $ \frac{a^{-2}}{a^{-5}} = a^{-2-(-5)} = a^{-2+5} = a^3 $. Just be careful with your negative signs.

Exponent Rules with Negative Powers

Simplify the following:

$\frac{a^4}{a^{-6}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's simplify this problem together.

00:10 Remember, when dividing powers with the same base,

00:14 the new power is the difference between the exponents.

00:18 So, we'll subtract the exponents to find the solution.

00:26 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following:

$\frac{a^4}{a^{-6}}=$

Step-by-step solution

Since a division operation between two terms with identical bases is required, we will use the power property to divide terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$ $\frac{c^m}{c^n}=c^{m-n}$ Note that using this property is only possible when the division is performed between terms with identical bases.

We return to the problem and apply the power property:

$\frac{a^4}{a^{-6}}=a^{4-(-6)}=a^{4+6}=a^{10}$ Therefore, the correct answer is option C.

Final Answer

$a^{10}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{a^4}{a^{-6}} = a^{4-(-6)} = a^{4+6} = a^{10}$
Check: Negative exponent in denominator becomes positive when moved up ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add 4 + (-6) = -2 to get a⁻²! Division means subtract the bottom exponent from the top exponent. Always use the rule: subtract the denominator exponent from the numerator exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I subtract a negative number?

When you subtract a negative number, it becomes addition! So $4 - (-6) = 4 + 6 = 10$ . Think of it as removing a debt - taking away negative 6 gives you positive 6.

What happens to the negative exponent in the denominator?

The negative exponent stays with the base when you apply the division rule. You're subtracting the entire exponent $-6$ , not just the 6.

Can I move the negative exponent to the numerator instead?

Yes! $\frac{a^4}{a^{-6}} = a^4 \times a^6 = a^{10}$ . Moving $a^{-6}$ up makes it $a^6$ , then multiply the powers.

How do I remember the division rule?

Think "top minus bottom" for division: $\frac{a^m}{a^n} = a^{m-n}$ . Always subtract the denominator exponent from the numerator exponent.

What if both exponents were negative?

Same rule applies! For $\frac{a^{-2}}{a^{-5}} = a^{-2-(-5)} = a^{-2+5} = a^3$ . Just be careful with your negative signs.

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