Solve for a: Simplifying (a³ᵇ)/(a²ᵇ) × aᵇ Using Exponent Rules

Q: Why do I subtract exponents when dividing but add when multiplying?

Think of it as opposite operations! Division undoes multiplication. When you multiply $ a^2 \cdot a^3 $, you're combining powers, so you add. When you divide $ \frac{a^5}{a^2} $, you're removing powers, so you subtract.

Q: What if the exponents are the same in division?

When exponents are identical like $ \frac{a^{2b}}{a^{2b}} $, you get $ a^{2b-2b} = a^0 = 1 $. Any number to the power of 0 equals 1!

Q: Can I work with the whole expression at once?

No, work step by step! First simplify $ \frac{a^{3b}}{a^{2b}} $ to get $ a^b $, then multiply by $ a^b $. Trying to do everything at once leads to errors.

Q: How do I remember which rule to use?

Use this memory trick: Division = Difference (subtract exponents), Multiplication = More (add exponents). Both start with the same letter!

Q: What if I get confused with the variable b in the exponent?

Treat $ b $ like any number! Whether it's $ a^3 $ or $ a^{3b} $, the exponent rules work the same way. Just do the arithmetic with whatever's in the exponent position.

Exponent Rules with Complex Multiplications

Solve for a:

$\frac{a^{3b}}{a^{2b}}\times a^b=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following expression

00:05 Move the product to the numerator

00:12 When multiplying powers with equal bases

00:16 The power of the result equals the sum of the exponents

00:21 We'll apply this formula to our exercise and then proceed to add the exponents

00:31 When dividing powers with equal bases

00:37 The power of the result equals the difference of the exponents

00:40 We'll apply this formula to our exercise and then subtract the exponents

00:46 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for a:

$\frac{a^{3b}}{a^{2b}}\times a^b=$

Step-by-step solution

Let's first deal with the first term in the multiplication, noting that the terms in the numerator and denominator have identical bases, so we'll use the power rule for division between terms with the same base:

$\frac{a^m}{a^n}=a^{m-n}$ We'll apply for the first term in the expression:

$\frac{a^{3b}}{a^{2b}}\cdot a^b=a^{3b-2b}\cdot a^b=a^b\cdot a^b$ where we also simplified the expression we got as a result of subtracting the exponents of the first term,

Next, we'll notice that the two terms in the multiplication have identical bases, so we'll use the power rule for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$ We'll apply to the problem:

$a^b\cdot a^b=a^{b+b}=a^{2b}$ Therefore, the correct answer is A.

Final Answer

$a^{2b}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: $\frac{a^m}{a^n} = a^{m-n}$ when bases are identical
Multiplication Rule: $a^m \cdot a^n = a^{m+n}$ for same bases: $a^b \cdot a^b = a^{2b}$
Check Order: Always simplify division first, then multiplication to get final answer ✓

Common Mistakes

Avoid these frequent errors

Adding exponents when dividing
Don't add exponents in $\frac{a^{3b}}{a^{2b}}$ = $a^{5b}$ ! This confuses multiplication with division rules. Always subtract exponents when dividing: $\frac{a^{3b}}{a^{2b}} = a^{3b-2b} = a^b$ .

Practice Quiz

Test your knowledge with interactive questions

Simplify the following equation:

$$ $4^5\times4^5=$

FAQ

Everything you need to know about this question

Why do I subtract exponents when dividing but add when multiplying?

Think of it as opposite operations! Division undoes multiplication. When you multiply $a^2 \cdot a^3$ , you're combining powers, so you add. When you divide $\frac{a^5}{a^2}$ , you're removing powers, so you subtract.

What if the exponents are the same in division?

When exponents are identical like $\frac{a^{2b}}{a^{2b}}$ , you get $a^{2b-2b} = a^0 = 1$ . Any number to the power of 0 equals 1!

Can I work with the whole expression at once?

No, work step by step! First simplify $\frac{a^{3b}}{a^{2b}}$ to get $a^b$ , then multiply by $a^b$ . Trying to do everything at once leads to errors.

How do I remember which rule to use?

Use this memory trick: Division = Difference (subtract exponents), Multiplication = More (add exponents). Both start with the same letter!

What if I get confused with the variable b in the exponent?

Treat $b$ like any number! Whether it's $a^3$ or $a^{3b}$ , the exponent rules work the same way. Just do the arithmetic with whatever's in the exponent position.

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Solve for a: Simplifying (a³ᵇ)/(a²ᵇ) × aᵇ Using Exponent Rules

Exponent Rules with Complex Multiplications

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I subtract exponents when dividing but add when multiplying?

What if the exponents are the same in division?

Can I work with the whole expression at once?

How do I remember which rule to use?

What if I get confused with the variable b in the exponent?

🌟 Unlock Your Math Potential

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