Solve: Simplifying (10×4)^(6+ax) ÷ (4×10)^(4+ax) Expression

Q: Why are (10×4) and (4×10) considered the same base?

Because multiplication is commutative, meaning the order doesn't matter! $ 10 \times 4 = 4 \times 10 = 40 $. They represent the exact same value, so they're the same base.

Q: What if the bases looked completely different?

If the bases were truly different (like $ 2^3 ÷ 3^2 $), you cannot use the quotient rule. The quotient rule $ a^m ÷ a^n = a^{m-n} $ only works when the bases are identical.

Q: How do I subtract the exponents correctly?

Write it as: $ (6+ax) - (4+ax) $. Distribute the negative: $ 6 + ax - 4 - ax $. The ax terms cancel out, leaving just $ 6 - 4 = 2 $.

Q: What happened to the 'ax' terms?

They canceled out during subtraction! $ +ax - ax = 0 $. This is why the final answer is simply $ (10 \times 4)^2 $, with no variable terms remaining.

Q: Can I simplify (10×4) to 40 in my final answer?

Yes, but it's not necessary! Both $ (10 \times 4)^2 $ and $ 40^2 $ are correct. The question asks for the expression form, so $ (10 \times 4)^2 $ matches the original format better.

Exponent Division with Identical Bases

Insert the corresponding expression:

$\frac{\left(10\times4\right)^{6+ax}}{\left(4\times10\right)^{4+ax}}=$

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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

00:06 We'll use this formula in our exercise and switch between the factors

00:13 We'll use the formula for dividing powers

00:15 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:18 equals the number (A) to the power of the difference of exponents (M-N)

00:21 We'll use this formula in our exercise

00:37 We'll open parentheses correctly

00:40 Negative times positive always equals negative

00:51 We'll group the factors

00:54 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:

$\frac{\left(10\times4\right)^{6+ax}}{\left(4\times10\right)^{4+ax}}=$

Step-by-step solution

To solve this problem, we need to simplify the given expression by applying the Power of a Quotient Rule for Exponents, which states that: $\frac{a^m}{a^n} = a^{m-n}$ .

We are given the expression:

$\frac{(10 \times 4)^{6+ax}}{(4 \times 10)^{4+ax}}$

Notice that $10 \times 4$ and $4 \times 10$ are equivalent, therefore:

$a = 10 \times 4$
$m = 6 + ax$
$n = 4 + ax$

Applying the quotient rule, we can write:

$\frac{a^m}{a^n} = a^{m-n}$

Inserting the values:

$a = (10 \times 4)$ , $m = 6 + ax$ , and $n = 4 + ax$ , we obtain:

$a^{6+ax-(4+ax)}$

By simplifying the exponents:

$(6 + ax) - (4 + ax) = 6 + ax - 4 - ax = 2$

Therefore, the expression becomes:

$(10 \times 4)^2$

The solution to the question is: $(10 \times 4)^2$

Final Answer

$\left(10\times4\right)^2$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with identical bases, subtract exponents
Technique: Recognize (10×4) = (4×10), then apply $a^m ÷ a^n = a^{m-n}$
Check: Simplify exponents first: (6+ax) - (4+ax) = 2 ✓

Common Mistakes

Avoid these frequent errors

Not recognizing that multiplication is commutative
Don't treat (10×4) and (4×10) as different bases = wrong answer! This prevents using the quotient rule for exponents. Always recognize that multiplication order doesn't matter: (10×4) = (4×10) = same base.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are (10×4) and (4×10) considered the same base?

Because multiplication is commutative, meaning the order doesn't matter! $10 \times 4 = 4 \times 10 = 40$ . They represent the exact same value, so they're the same base.

What if the bases looked completely different?

If the bases were truly different (like $2^3 ÷ 3^2$ ), you cannot use the quotient rule. The quotient rule $a^m ÷ a^n = a^{m-n}$ only works when the bases are identical.

How do I subtract the exponents correctly?

Write it as: $(6+ax) - (4+ax)$ . Distribute the negative: $6 + ax - 4 - ax$ . The ax terms cancel out, leaving just $6 - 4 = 2$ .

What happened to the 'ax' terms?

They canceled out during subtraction! $+ax - ax = 0$ . This is why the final answer is simply $(10 \times 4)^2$ , with no variable terms remaining.

Can I simplify (10×4) to 40 in my final answer?

Yes, but it's not necessary! Both $(10 \times 4)^2$ and $40^2$ are correct. The question asks for the expression form, so $(10 \times 4)^2$ matches the original format better.

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