Simplify (3×6)¹⁰ ÷ (3×6)⁷: Division of Exponents Practice

Q: Why do we subtract exponents when dividing?

Think of it as canceling out common factors! When you have $ \frac{a^{10}}{a^7} $, you can cancel 7 copies of 'a' from both top and bottom, leaving $ a^{10-7} = a^3 $.

Q: What if the bases aren't exactly the same?

The bases must be identical to use this rule! In our problem, both numerator and denominator have the base $ (3×6) $, so we can apply the rule.

Q: Do I need to calculate 3×6 first?

No! Keep the base as $ (3×6) $ throughout the problem. The rule works regardless of what the base equals - focus on the exponents.

Q: What happens if the bottom exponent is larger?

You still subtract! For example, $ \frac{a^3}{a^5} = a^{3-5} = a^{-2} $. Negative exponents are perfectly valid and mean $ \frac{1}{a^2} $.

Q: How can I remember this rule?

Think: "Same base, subtract the powers!" Division means taking away, so you subtract exponents. Multiplication would mean adding them together.

Exponent Division with Same Base

Insert the corresponding expression:

$\frac{\left(3\times6\right)^{10}}{\left(3\times6\right)^7}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's get started.

00:14 We'll use the formula for dividing powers.

00:18 If you have a number A raised to the power of N, divided by the same number A raised to the power of M,

00:24 it equals the number A raised to the power of M minus N.

00:29 Let's apply this formula to our exercise now.

00:33 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{\left(3\times6\right)^{10}}{\left(3\times6\right)^7}=$

Step-by-step solution

We need to simplify the expression: $\frac{\left(3\times6\right)^{10}}{\left(3\times6\right)^7}$ .

According to the Power of a Quotient Rule for Exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$ , we can simplify any fraction where the numerator and the denominator have the same base and different exponents by subtracting their exponents.

In our case, the common base is $3\times6$ . Let's apply the rule:

The exponent in the numerator is 10.
The exponent in the denominator is 7.

So, according to the rule, we subtract the exponent in the denominator from the exponent in the numerator:

$(3\times6)^{10-7}$ .

Thus, the expression simplifies to $\left(3\times6\right)^{10-7}$ .

Final Answer

$\left(3\times6\right)^{10-7}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract the exponents
Technique: $\frac{a^m}{a^n} = a^{m-n}$ means 10 - 7 = 3
Check: $(3×6)^3 = 18^3$ equals the original expression ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add (10 + 7 = 17) or multiply (10 × 7 = 70) the exponents = completely wrong answer! Division requires subtraction because you're canceling out common factors. 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 it as canceling out common factors! When you have $\frac{a^{10}}{a^7}$ , you can cancel 7 copies of 'a' from both top and bottom, leaving $a^{10-7} = a^3$ .

What if the bases aren't exactly the same?

The bases must be identical to use this rule! In our problem, both numerator and denominator have the base $(3×6)$ , so we can apply the rule.

Do I need to calculate 3×6 first?

No! Keep the base as $(3×6)$ throughout the problem. The rule works regardless of what the base equals - focus on the exponents.

What happens if the bottom exponent is larger?

You still subtract! For example, $\frac{a^3}{a^5} = a^{3-5} = a^{-2}$ . Negative exponents are perfectly valid and mean $\frac{1}{a^2}$ .

How can I remember this rule?

Think: "Same base, subtract the powers!" Division means taking away, so you subtract exponents. Multiplication would mean adding them together.

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