Simplify (11×12)^30 ÷ (11×12)^30: Power Division Problem

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{a^3}{a^2} = \frac{a \times a \times a}{a \times a} $. Two a's cancel out, leaving just one a, which is $ a^1 = a^{3-2} $!

Q: What happens when the exponents are the same?

When exponents are equal, like $ \frac{(11 \times 12)^{30}}{(11 \times 12)^{30}} $, you get $ (11 \times 12)^{30-30} = (11 \times 12)^0 = 1 $. Any number divided by itself equals 1!

Q: Do I need to calculate 11 × 12 first?

No! Keep it as $ (11 \times 12) $ since it's the same base in both numerator and denominator. The actual value doesn't matter for applying the quotient rule.

Q: What if the bases were different numbers?

The quotient rule only works when bases are identical. If you had $ \frac{11^{30}}{12^{30}} $, you couldn't subtract exponents because the bases (11 and 12) are different.

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

1

Understand the problem

Insert the corresponding expression:

$\frac{\left(11\times12\right)^{30}}{\left(11\times12\right)^{30}}=$

2

Step-by-step solution

Let's solve the given mathematical expression step by step using the rules of exponents.

We start with the expression: $\frac{\left(11\times12\right)^{30}}{\left(11\times12\right)^{30}}$ .
According to the rules of exponents, specifically the quotient rule, which states that when you divide powers with the same base, you subtract their exponents: $\frac{a^m}{a^n} = a^{m-n}$ .
Applying this rule to the expression, since the base $11 \times 12$ is the same in both the numerator and the denominator, we subtract the exponents:
- The numerator is $\left(11\times12\right)^{30}$ and the denominator is $\left(11\times12\right)^{30}$ .
- Therefore, $\frac{\left(11\times12\right)^{30}}{\left(11\times12\right)^{30}} = \left(11\times12\right)^{30-30}$ .
Simplifying further, we have:
- $\left(11\times12\right)^{30-30} = \left(11\times12\right)^{0}$ .
- Any non-zero number raised to the power of 0 is 1. However, here the expression is left in the form of an exponent as requested.

The solution to the question is: $\left(11\times12\right)^{30-30}$

3

Final Answer

$\left(11\times12\right)^{30-30}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract the exponents
Technique: $\frac{a^m}{a^n} = a^{m-n}$ means 30 - 30
Check: Any non-zero number to the zero power equals 1 ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents (30 + 30) or multiply them (30 × 30) when dividing = completely wrong operation! This confuses division with multiplication or addition rules. Always subtract exponents when dividing powers with the same base.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

+

Think of it this way: $\frac{a^3}{a^2} = \frac{a \times a \times a}{a \times a}$ . Two a's cancel out, leaving just one a, which is $a^1 = a^{3-2}$ !

What happens when the exponents are the same?

+

When exponents are equal, like $\frac{(11 \times 12)^{30}}{(11 \times 12)^{30}}$ , you get $(11 \times 12)^{30-30} = (11 \times 12)^0 = 1$ . Any number divided by itself equals 1!

Do I need to calculate 11 × 12 first?

+

No! Keep it as $(11 \times 12)$ since it's the same base in both numerator and denominator. The actual value doesn't matter for applying the quotient rule.

What if the bases were different numbers?

+

The quotient rule only works when bases are identical. If you had $\frac{11^{30}}{12^{30}}$ , you couldn't subtract exponents because the bases (11 and 12) are different.

Is the answer always 1 when exponents are equal?

+

Yes! When you divide any non-zero number by itself, you always get 1. So $\frac{a^n}{a^n} = a^{n-n} = a^0 = 1$ for any non-zero base a.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Simplify (11×12)^30 ÷ (11×12)^30: Power Division Problem

Exponent Division with Same Bases

❤️ Continue Your Math Journey!