Simplify the Expression: b²/b¹ Using Exponent Rules

Q: Why do we subtract exponents when dividing?

Think of exponents as counting factors! $ b^2 = b \cdot b $ and $ b^1 = b $. When you divide $ \frac{b \cdot b}{b} $, you cancel one b from top and bottom, leaving just one b = $ b^1 $.

Q: What if the exponent in the denominator is bigger?

You still subtract! For example, $ \frac{b^2}{b^5} = b^{2-5} = b^{-3} $. The negative exponent means the result is a fraction: $ \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. $ \frac{a^2}{b^1} $ cannot be simplified using this rule because a and b are different.

Q: Is there a shortcut for this type of problem?

Yes! When you see $ \frac{b^m}{b^n} $, immediately write $ b^{m-n} $. For $ \frac{b^2}{b^1} $, think "2 minus 1 equals 1" so the answer is $ b^1 $.

Quotient Rule with Same Base Exponents

Insert the corresponding expression:

$\frac{b^2}{b^1}=$

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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)

00:07 equals the number (A) to the power of the difference of exponents (M-N)

00:10 We'll use this formula in our exercise

00:12 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{b^2}{b^1}=$

Step-by-step solution

To solve the problem, we apply the quotient rule for exponents. The quotient rule states that when you divide two exponential expressions with the same base, you can subtract the exponent of the denominator from the exponent of the numerator. In mathematical terms:

$\frac{a^m}{a^n} = a^{m-n}$

Using the formula mentioned above, let's solve $\frac{b^2}{b^1}$ :

Identify the base for both the numerator and the denominator, which in this case is $b$
Identify the exponents for the numerator and the denominator: $m = 2$ and $n = 1$
Subtract the exponent in the denominator from the exponent in the numerator: $2 - 1 = 1$
Rewrite the expression with the new exponent: $b^{2-1} = b^1$

Thus, $\frac{b^2}{b^1} = b^1$ as per the power of a quotient rule.

The solution to the question is: $b^1$

Final Answer

$b^1$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract exponents: $a^m ÷ a^n = a^{m-n}$
Technique: Identify exponents then subtract: $b^2 ÷ b^1 = b^{2-1} = b^1$
Check: Expand to verify: $\frac{b \cdot b}{b} = b$ which equals $b^1$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add the exponents like $b^2 ÷ b^1 = b^{2+1} = b^3$ ! This is the multiplication rule, not division. Division means you have fewer factors, so the exponent gets smaller. Always subtract the bottom exponent from the top exponent.

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 exponents as counting factors! $b^2 = b \cdot b$ and $b^1 = b$ . When you divide $\frac{b \cdot b}{b}$ , you cancel one b from top and bottom, leaving just one b = $b^1$ .

What if the exponent in the denominator is bigger?

You still subtract! For example, $\frac{b^2}{b^5} = b^{2-5} = b^{-3}$ . The negative exponent means the result is a fraction: $\frac{1}{b^3}$ .

Can I use this rule with different bases?

No! The quotient rule only works when the bases are exactly the same. $\frac{a^2}{b^1}$ cannot be simplified using this rule because a and b are different.

What does b¹ actually mean?

Any number or variable raised to the power of 1 equals itself! So $b^1 = b$ , $5^1 = 5$ , etc. It's like saying "one group of b" = just b.

Is there a shortcut for this type of problem?

Yes! When you see $\frac{b^m}{b^n}$ , immediately write $b^{m-n}$ . For $\frac{b^2}{b^1}$ , think "2 minus 1 equals 1" so the answer is $b^1$ .

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