Simplify the Expression: Division of Powers b^11 ÷ b^8

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{b^{11}}{b^8} $ means 11 b's multiplied together divided by 8 b's multiplied together. The 8 b's cancel out, leaving you with 11 - 8 = 3 b's, so $ b^3 $!

Q: What if the bottom exponent is bigger than the top?

You still subtract! For example, $ \frac{b^5}{b^8} = b^{5-8} = b^{-3} $. The negative exponent means one divided by that positive power: $ \frac{1}{b^3} $.

Q: Can I use this rule with different bases?

No! The quotient rule only works when the bases are exactly the same. For $ \frac{a^5}{b^3} $, you cannot subtract exponents because a and b are different.

Q: How do I remember which operation to use?

Remember: Multiplication = Add exponents, Division = Subtract exponents. Think of division as the opposite of multiplication, so you do the opposite operation with exponents too!

Q: What happens if both exponents are the same?

Great question! $ \frac{b^8}{b^8} = b^{8-8} = b^0 = 1 $. Any number (except zero) raised to the power of 0 equals 1. This makes sense because anything divided by itself equals 1!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 We'll use the formula for dividing powers

00:04 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:06 equals the number (A) to the power of the difference of exponents (M-N)

00:08 We'll use this formula in our exercise

00:10 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the corresponding expression:

$\frac{b^{11}}{b^8}=$

2

Step-by-step solution

To solve this problem, we will apply the Quotient Rule for exponents, which helps simplify expressions where both the numerator and denominator share the same base.

Step 1: Identify the given information.
We are given the expression $\frac{b^{11}}{b^8}$ .
Step 2: Apply the Quotient Rule for exponents.
According to the Quotient Rule, $\frac{b^m}{b^n} = b^{m-n}$ . Here, $m = 11$ and $n = 8$ .
Step 3: Subtract the exponent of the denominator from the exponent of the numerator.
Calculate $11 - 8 = 3$ , leading to the simplified expression $b^3$ .

Therefore, the simplified form of the given expression is $b^{3}$ .

When considering the choices:

Choice 1: $b^{11+8}$ adds the exponents, which is incorrect for division.
Choice 2: $b^{11\times8}$ multiplies the exponents, also incorrect for division.
Choice 3: $b^{\frac{11}{8}}$ divides the exponents, which is incorrect for division.
Choice 4: $b^{11-8}$ , correctly subtracts the exponents as per the Quotient Rule.

Thus, the correct choice is Choice 4: $b^{11-8}$ , which simplifies to $b^3$ .

3

Final Answer

$b^{11-8}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract the exponents
Technique: $\frac{b^{11}}{b^8} = b^{11-8} = b^3$
Check: Multiply result by denominator: $b^3 \times b^8 = b^{11}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add exponents like $b^{11+8} = b^{19}$ when dividing! This gives a massive wrong answer because addition is for multiplication, not division. Always subtract the bottom exponent from the top exponent when dividing same bases.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

+

Think of it this way: $\frac{b^{11}}{b^8}$ means 11 b's multiplied together divided by 8 b's multiplied together. The 8 b's cancel out, leaving you with 11 - 8 = 3 b's, so $b^3$ !

What if the bottom exponent is bigger than the top?

+

You still subtract! For example, $\frac{b^5}{b^8} = b^{5-8} = b^{-3}$ . The negative exponent means one divided by that positive power: $\frac{1}{b^3}$ .

Can I use this rule with different bases?

+

No! The quotient rule only works when the bases are exactly the same. For $\frac{a^5}{b^3}$ , you cannot subtract exponents because a and b are different.

How do I remember which operation to use?

+

Remember: Multiplication = Add exponents, Division = Subtract exponents. Think of division as the opposite of multiplication, so you do the opposite operation with exponents too!

What happens if both exponents are the same?

+

Great question! $\frac{b^8}{b^8} = b^{8-8} = b^0 = 1$ . Any number (except zero) raised to the power of 0 equals 1. This makes sense because anything divided by itself equals 1!

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Simplify the Expression: Division of Powers b^11 ÷ b^8

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