Simplify the Expression: 12b⁴ ÷ 4b⁻⁵ Using Power Rules

Q: How do I remember the division rule for exponents?

Remember: "Same base? Subtract!" For $ \frac{a^m}{a^n} = a^{m-n} $. The top exponent minus the bottom exponent. It's the opposite of multiplication where you add exponents.

Q: What if I get confused with the negative exponent?

Try rewriting first! $ b^{-5} = \frac{1}{b^5} $, so $ \frac{12b^4}{4b^{-5}} = \frac{12b^4}{\frac{4}{b^5}} = \frac{12b^4 \cdot b^5}{4} = 3b^9 $. Same answer, different path!

Q: Can I check my answer by plugging in a number?

Yes! Try b = 2. Original: $ \frac{12(2)^4}{4(2)^{-5}} = \frac{12(16)}{4(\frac{1}{32})} = \frac{192}{\frac{1}{8}} = 1536 $. Answer: $ 3(2)^9 = 3(512) = 1536 $ ✓

Exponent Division with Negative Powers

Complete the exercise:

$\frac{12b^4}{4b^{-5}}=$

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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 When dividing powers with equal bases

00:07 The power of the result equals the difference between the powers

00:11 We'll apply this formula to our exercise, and subtract the powers

00:21 Let's calculate 12 divided by 4

00:29 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the exercise:

$\frac{12b^4}{4b^{-5}}=$

Step-by-step solution

Let's consider that the numerator and the denominator of the fraction have terms with identical bases, therefore we will use the law of exponents for the division of terms with identical bases:

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

$\frac{12b^4}{4b^{-5}}=3\cdot b^{4-(-5)}=3\cdot b^{4+5}=3b^9$ When in the first step we simplify the numerical part of the fraction. This operation is intuitive as well as correct since it is possible to write down in advance the said fraction as a product of fractions and reduce:

$\frac{12b^4}{4b^{-5}}=\frac{12}{4}\cdot\frac{b^4}{b^{-5}}=3\cdot b^{4-(-5)}=\ldots$ We return once again to the problem. The simplified expression obtained is as follows:

$3b^9$

Therefore, the correct answer is option D.

Final Answer

$3b^9$

Key Points to Remember

Essential concepts to master this topic

Division Rule: When dividing powers with same base: subtract exponents
Technique: With negative exponents: $4 - (-5) = 4 + 5 = 9$
Check: Verify by converting: $\frac{12b^4}{4b^{-5}} = \frac{12b^4 \cdot b^5}{4} = 3b^9$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting when dividing
Don't add exponents when dividing: 4 + (-5) = -1 gives wrong answer $3b^{-1}$ ! Division requires subtracting the bottom exponent from the top. Always subtract: 4 - (-5) = 4 + 5 = 9.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does subtracting a negative number become addition?

When you subtract a negative, it's like removing a debt, which adds value! So $4 - (-5)$ becomes $4 + 5 = 9$ . Think: "minus negative equals plus positive."

What happens to the coefficients 12 and 4?

The coefficients are regular numbers, so just divide them normally: $\frac{12}{4} = 3$ . Handle the numbers and variables separately, then combine the results.

How do I remember the division rule for exponents?

Remember: "Same base? Subtract!" For $\frac{a^m}{a^n} = a^{m-n}$ . The top exponent minus the bottom exponent. It's the opposite of multiplication where you add exponents.

What if I get confused with the negative exponent?

Try rewriting first! $b^{-5} = \frac{1}{b^5}$ , so $\frac{12b^4}{4b^{-5}} = \frac{12b^4}{\frac{4}{b^5}} = \frac{12b^4 \cdot b^5}{4} = 3b^9$ . Same answer, different path!

Can I check my answer by plugging in a number?

Yes! Try b = 2. Original: $\frac{12(2)^4}{4(2)^{-5}} = \frac{12(16)}{4(\frac{1}{32})} = \frac{192}{\frac{1}{8}} = 1536$ . Answer: $3(2)^9 = 3(512) = 1536$ ✓

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