Simplify (4×7)^(2x)/(4×7)^4: Exponential Expression with Variable Power

Q: Why do I subtract the exponents instead of dividing them?

The Quotient Rule for Exponents says $ \frac{a^m}{a^n} = a^{m-n} $. Dividing powers with the same base means we subtract exponents, not divide them!

Q: Does the order matter when subtracting exponents?

Yes, absolutely! Always subtract the denominator exponent from the numerator exponent. Here: $ 2x - 4 $, not $ 4 - 2x $.

Q: What happens to the base (4×7)?

The base stays exactly the same! We only work with the exponents when using the quotient rule. The base $ (4×7) $ appears in both our original expression and final answer.

Q: Can I simplify (4×7) to 28 first?

You could, but it's not necessary! The question asks for the expression form, so keeping $ (4×7) $ as the base is perfectly fine and matches the answer choices.

Q: How do I know which answer choice is correct?

Look for the one with the correct exponent: $ 2x - 4 $. Remember, when dividing powers, you subtract exponents in the order: numerator minus denominator.

Quotient Rule with Variable Exponents

Insert the corresponding expression:

$\frac{\left(4\times7\right)^{2x}}{\left(4\times7\right)^4}=$

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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 exponents

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

Understand the problem

Insert the corresponding expression:

$\frac{\left(4\times7\right)^{2x}}{\left(4\times7\right)^4}=$

Step-by-step solution

We start with the expression: $\frac{\left(4\times7\right)^{2x}}{\left(4\times7\right)^4}$ .

According to the Power of a Quotient Rule for Exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$ , we can simplify the expression by subtracting the exponents.

The base here is $4 \times 7$ , and it is common in both the numerator and the denominator.

Thus, using the exponent rule, we have:

Exponent in the numerator: $2x$
Exponent in the denominator: $4$

Now, apply the rule:

$\frac{\left(4\times7\right)^{2x}}{\left(4\times7\right)^4} = \left(4\times7\right)^{2x-4}$

The solution to the question is: $\left(4\times7\right)^{2x-4}$ .

Final Answer

$\left(4\times7\right)^{2x-4}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{a^m}{a^n} = a^{m-n}$ gives us $2x - 4$
Check: Base $(4×7)$ stays same, exponent becomes $2x-4$ ✓

Common Mistakes

Avoid these frequent errors

Subtracting exponents in wrong order
Don't subtract $4 - 2x$ = wrong answer! This gives $(4×7)^{4-2x}$ instead of the correct result. Always subtract denominator exponent from numerator exponent: $2x - 4$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I subtract the exponents instead of dividing them?

The Quotient Rule for Exponents says $\frac{a^m}{a^n} = a^{m-n}$ . Dividing powers with the same base means we subtract exponents, not divide them!

Does the order matter when subtracting exponents?

Yes, absolutely! Always subtract the denominator exponent from the numerator exponent. Here: $2x - 4$ , not $4 - 2x$ .

What happens to the base (4×7)?

The base stays exactly the same! We only work with the exponents when using the quotient rule. The base $(4×7)$ appears in both our original expression and final answer.

Can I simplify (4×7) to 28 first?

You could, but it's not necessary! The question asks for the expression form, so keeping $(4×7)$ as the base is perfectly fine and matches the answer choices.

How do I know which answer choice is correct?

Look for the one with the correct exponent: $2x - 4$ . Remember, when dividing powers, you subtract exponents in the order: numerator minus denominator.

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