Solve (12×2)⁵ Divided by (2×12)^(3y): Exponential Equation Challenge

Q: Why are (12×2) and (2×12) considered the same base?

Because multiplication is commutative! This means $ 12 \times 2 = 2 \times 12 = 24 $. The order doesn't matter, so they represent the same base value.

Q: What if the bases looked completely different?

If the bases were truly different (like $ \frac{3^5}{4^{3y}} $), you cannot use the quotient rule. The quotient rule only works when bases are identical.

Q: Why do we subtract 3y from 5 instead of adding?

The quotient rule states $ \frac{a^m}{a^n} = a^{m-n} $. Since we're dividing, we subtract the bottom exponent from the top exponent: 5 - 3y.

Q: What if 5-3y equals zero?

If $ 5 - 3y = 0 $, then our answer becomes $ (12 \times 2)^0 = 1 $. Any non-zero number raised to the power of 0 equals 1!

Q: Can I calculate 12×2 first to make it easier?

Yes! You could rewrite this as $ \frac{24^5}{24^{3y}} = 24^{5-3y} $. The answer is the same, just with a simplified base.

Exponential Quotients with Same Base

Insert the corresponding expression:

$\frac{\left(12\times2\right)^5}{\left(2\times12\right)^{3y}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 The order of factors in multiplication doesn't matter

00:08 We'll use this formula in our exercise and reverse the order of factors

00:15 We'll use the formula for dividing powers

00:17 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:20 equals the number (A) to the power of the difference of exponents (M-N)

00:23 We'll use this formula in our exercise

00:26 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{\left(12\times2\right)^5}{\left(2\times12\right)^{3y}}=$

Step-by-step solution

To solve the given expression $\frac{\left(12\times2\right)^5}{\left(2\times12\right)^{3y}}$ , we need to apply the rule for the power of a quotient for exponents: $\frac{a^m}{a^n} = a^{m-n}$ .

The expressions in both the numerator and the denominator have the same base $\left(12 \times 2\right)$ . Therefore, the expression can be rewritten as:

Base: $\left(12 \times 2\right)$
Exponent in the numerator: $5$
Exponent in the denominator: $3y$

Now, applying the quotient rule:

$\frac{\left(12\times2\right)^5}{\left(2\times12\right)^{3y}} = \left(12\times2\right)^{5-3y}$

The solution to the question is:

$\left(12\times2\right)^{5-3y}$

Final Answer

$\left(12\times2\right)^{5-3y}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{a^m}{a^n} = a^{m-n}$ so 5 - 3y in exponent
Check: Verify base is identical: (12×2) = (2×12) = 24 ✓

Common Mistakes

Avoid these frequent errors

Not recognizing that (12×2) and (2×12) are the same base
Don't treat (12×2) and (2×12) as different bases = can't apply quotient rule! Multiplication is commutative, so 12×2 = 2×12 = 24. Always simplify or recognize equivalent expressions as the same base.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are (12×2) and (2×12) considered the same base?

Because multiplication is commutative! This means $12 \times 2 = 2 \times 12 = 24$ . The order doesn't matter, so they represent the same base value.

What if the bases looked completely different?

If the bases were truly different (like $\frac{3^5}{4^{3y}}$ ), you cannot use the quotient rule. The quotient rule only works when bases are identical.

Why do we subtract 3y from 5 instead of adding?

The quotient rule states $\frac{a^m}{a^n} = a^{m-n}$ . Since we're dividing, we subtract the bottom exponent from the top exponent: 5 - 3y.

What if 5-3y equals zero?

If $5 - 3y = 0$ , then our answer becomes $(12 \times 2)^0 = 1$ . Any non-zero number raised to the power of 0 equals 1!

Can I calculate 12×2 first to make it easier?

Yes! You could rewrite this as $\frac{24^5}{24^{3y}} = 24^{5-3y}$ . The answer is the same, just with a simplified base.

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