Simplify the Expression: b⁵/b² Using Exponent Rules

Q: Why do I subtract exponents when dividing?

Think about what division means! $ \frac{b^5}{b^2} $ means how many times does $ b^2 $ go into $ b^5 $? Since $ b^2 \cdot b^3 = b^5 $, the answer is $ b^3 $.

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

You still subtract! For example, $ \frac{b^2}{b^5} = b^{2-5} = b^{-3} $. The negative exponent means one over that positive power: $ b^{-3} = \frac{1}{b^3} $.

Q: Can I cancel out the b's like regular fractions?

Yes! You can think of it as canceling: $ \frac{b \cdot b \cdot b \cdot b \cdot b}{b \cdot b} $ - cancel two b's from top and bottom, leaving $ b \cdot b \cdot b = b^3 $.

Q: Do I need the bases to be exactly the same?

Yes! This rule only works when the bases are identical. You cannot simplify $ \frac{a^5}{b^2} $ using exponent rules because the bases are different.

Q: What if there's a coefficient like 3b⁵/b²?

Handle the coefficient separately! $ \frac{3b^5}{b^2} = 3 \cdot \frac{b^5}{b^2} = 3 \cdot b^3 = 3b^3 $. The exponent rule only applies to the variable parts.

Exponent Division with Same Base

Insert the corresponding expression:

$\frac{b^5}{b^2}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's start by understanding the process.

00:10 We use a formula for dividing powers.

00:14 When dividing powers with the same base, like A,

00:18 you raise A to the power of M minus N, which is the difference of the exponents.

00:25 Let's use this rule in our exercise.

00:28 We'll compare the numbers to the variables, keep the base, and subtract the exponents.

00:34 Doing this gives us the solution to the problem. Great job!

Step-by-step written solution

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Understand the problem

Insert the corresponding expression:

$\frac{b^5}{b^2}=$

Step-by-step solution

To solve this problem, we need to simplify the expression $\frac{b^5}{b^2}$ using the rules of exponents.

Step 1: Identify the rule to apply: For any positive integer exponents $m$ and $n$ , the rule $\frac{a^m}{a^n} = a^{m-n}$ applies when dividing terms with the same base. In this expression, our base is $b$ .
Step 2: Apply the rule: Substitute the given exponents into the formula: $\frac{b^5}{b^2} = b^{5-2}$
Step 3: Perform the subtraction: Calculate the exponent $5 - 2$ : $b^{5-2} = b^3$

Therefore, the solution to the expression $\frac{b^5}{b^2}$ is $b^3$ .

Final Answer

$b^3$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: Apply formula: $\frac{b^5}{b^2} = b^{5-2} = b^3$
Check: Verify by expanding: $\frac{b \cdot b \cdot b \cdot b \cdot b}{b \cdot b} = b^3$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add 5 + 2 = 7 to get $b^7$ ! Addition is for multiplication, not division. This gives completely wrong answers. 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 I subtract exponents when dividing?

Think about what division means! $\frac{b^5}{b^2}$ means how many times does $b^2$ go into $b^5$ ? Since $b^2 \cdot b^3 = b^5$ , the answer is $b^3$ .

What if the bottom exponent is bigger than the top?

You still subtract! For example, $\frac{b^2}{b^5} = b^{2-5} = b^{-3}$ . The negative exponent means one over that positive power: $b^{-3} = \frac{1}{b^3}$ .

Can I cancel out the b's like regular fractions?

Yes! You can think of it as canceling: $\frac{b \cdot b \cdot b \cdot b \cdot b}{b \cdot b}$ - cancel two b's from top and bottom, leaving $b \cdot b \cdot b = b^3$ .

Do I need the bases to be exactly the same?

Yes! This rule only works when the bases are identical. You cannot simplify $\frac{a^5}{b^2}$ using exponent rules because the bases are different.

What if there's a coefficient like 3b⁵/b²?

Handle the coefficient separately! $\frac{3b^5}{b^2} = 3 \cdot \frac{b^5}{b^2} = 3 \cdot b^3 = 3b^3$ . The exponent rule only applies to the variable parts.

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