Simplify Complex Expression: [a⁴/a³ × a⁸/a⁷] ÷ a¹⁰/a⁸

Q: Why do I subtract exponents when dividing?

Think of it as canceling out repeated multiplication! $ \frac{a^4}{a^3} = \frac{a \times a \times a \times a}{a \times a \times a} $ - three a's cancel out, leaving just one a.

Q: What happens when I get a^0 as my answer?

Don't panic! $ a^0 = 1 $ for any non-zero value of a. This is a fundamental rule in mathematics - it's like saying "no factors of a remain."

Q: How do I handle division by a fraction like in this problem?

Dividing by a fraction is the same as multiplying by its reciprocal. So $ a^2 ÷ \frac{a^{10}}{a^8} = a^2 \times \frac{a^8}{a^{10}} $

Q: Should I work left to right or follow some other order?

Follow the order of operations! First simplify what's in brackets, then handle the division. This prevents errors and keeps your work organized.

Q: Can I get a negative exponent in my final answer?

Yes! If you get $ a^{-n} $, it equals $ \frac{1}{a^n} $. But in this problem, the negative exponents cancel out to give us $ a^0 = 1 $.

Exponent Rules with Division Operations

Simplify the following:

$\lbrack\frac{a^4}{a^3}\times\frac{a^8}{a^7}\rbrack:\frac{a^{10}}{a^8}$

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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

00:06 The power of the result equals the difference of the exponents

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

00:36 When multiplying powers with equal bases

00:41 The power of the result equals the sum of the exponents

00:44 We'll apply this formula to our exercise, and add the exponents together

00:55 Any number divided by itself always equals 1

01:00 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following:

$\lbrack\frac{a^4}{a^3}\times\frac{a^8}{a^7}\rbrack:\frac{a^{10}}{a^8}$

Step-by-step solution

To solve this problem, we need to simplify the given expression using the rules of exponents:

First, simplify inside the brackets:
$\frac{a^4}{a^3} \times \frac{a^8}{a^7} = a^{4-3} \times a^{8-7} = a^1 \times a^1 = a^{1+1} = a^2$

Now, handle the entire expression, dividing it by $\frac{a^{10}}{a^8}$ :
$\frac{a^2}{\frac{a^{10}}{a^8}} = a^2 \times \frac{a^8}{a^{10}} = a^2 \times a^{8-10} = a^2 \times a^{-2} = a^{2 + (-2)} = a^0$

Recall that any non-zero number raised to the power of zero is 1, hence: $a^0 = 1$

Therefore, the solution to the problem is $1$ .

Final Answer

$1$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing powers, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: Simplify step by step: $\frac{a^4}{a^3} \times \frac{a^8}{a^7} = a^1 \times a^1 = a^2$
Check: Any non-zero base raised to power zero equals 1: $a^0 = 1$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents when dividing
Don't add exponents when you see division = $\frac{a^4}{a^3} = a^{4+3} = a^7$ ! This gives completely wrong results because you're applying multiplication rules to division. Always subtract the bottom exponent from the top exponent when dividing.

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?

Think of it as canceling out repeated multiplication! $\frac{a^4}{a^3} = \frac{a \times a \times a \times a}{a \times a \times a}$ - three a's cancel out, leaving just one a.

What happens when I get a^0 as my answer?

Don't panic! $a^0 = 1$ for any non-zero value of a. This is a fundamental rule in mathematics - it's like saying "no factors of a remain."

How do I handle division by a fraction like in this problem?

Dividing by a fraction is the same as multiplying by its reciprocal. So $a^2 ÷ \frac{a^{10}}{a^8} = a^2 \times \frac{a^8}{a^{10}}$

Should I work left to right or follow some other order?

Follow the order of operations! First simplify what's in brackets, then handle the division. This prevents errors and keeps your work organized.

Can I get a negative exponent in my final answer?

Yes! If you get $a^{-n}$ , it equals $\frac{1}{a^n}$ . But in this problem, the negative exponents cancel out to give us $a^0 = 1$ .

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Simplify Complex Expression: [a⁴/a³ × a⁸/a⁷] ÷ a¹⁰/a⁸

Exponent Rules with Division Operations

❤️ 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?

What happens when I get a^0 as my answer?

How do I handle division by a fraction like in this problem?

Should I work left to right or follow some other order?

Can I get a negative exponent in my final answer?

🌟 Unlock Your Math Potential

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