Simplify (7×2)^9 ÷ (2×7)^2: Expression with Different Exponents

Q: Why are (7×2) and (2×7) considered the same base?

Because multiplication is commutative, meaning $ 7×2 = 2×7 = 14 $. The order doesn't matter - both equal the same value, so they're the same base!

Q: When do I subtract exponents versus add them?

Subtract when dividing: $ \frac{a^m}{a^n} = a^{m-n} $Add when multiplying: $ a^m × a^n = a^{m+n} $Think: division reduces the power!

Q: What if the exponent in the denominator is larger?

You still subtract! If you get a negative exponent, that's correct. For example: $ \frac{x^2}{x^5} = x^{2-5} = x^{-3} $

Q: Do I need to calculate (7×2) first?

No! Keep it as $ (7×2) $ in your answer. The question asks for the expression form, not the final numerical value.

Q: How can I remember the subtraction rule?

Think of it this way: division cancels out multiplication. If you have $ \frac{a×a×a}{a×a} $, you cancel 2 a's from top and bottom, leaving one a. That's $ a^{3-2} = a^1 $!

Exponent Division with Same Base

Insert the corresponding expression:

$\frac{\left(7\times2\right)^{9}}{\left(2\times7\right)^2}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 In multiplication, the order of factors doesn't matter

00:07 We'll use this formula in our exercise and reverse the order of factors

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

00:18 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:21 equals the number (A) to the power of the difference of exponents (M-N)

00:24 We'll use this formula in our exercise

00:27 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:

$\frac{\left(7\times2\right)^{9}}{\left(2\times7\right)^2}=$

Step-by-step solution

To solve the given expression $\frac{\left(7\times2\right)^{9}}{\left(2\times7\right)^2}$ , we will apply the Power of a Quotient Rule for Exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$ .

The base of the exponents in both the numerator and the denominator is the same, $7 \times 2$ or equivalently $2 \times 7$ .

1. First, note that the structure is $\frac{(7\times2)^9}{(7\times2)^2}$ .

2. Using the Power of a Quotient Rule: $\frac{(7\times2)^9}{(7\times2)^2} = (7\times2)^{9-2}$

3. Simplify the expression in the exponent: $9 - 2 = 7$

4. Therefore, the simplified expression is \ $(7\times2)^7$

The solution to the question is: $(7\times2)^{9-2}$

Final Answer

$\left(7\times2\right)^{9-2}$

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 $\frac{(7×2)^9}{(7×2)^2} = (7×2)^{9-2}$
Check: Verify that $7×2 = 2×7 = 14$ makes bases identical ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add the exponents 9 + 2 = 11 when dividing = $(7×2)^{11}$ ! This gives a completely wrong answer because you're multiplying instead of dividing. Always subtract exponents when dividing: $\frac{a^m}{a^n} = a^{m-n}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are (7×2) and (2×7) considered the same base?

Because multiplication is commutative, meaning $7×2 = 2×7 = 14$ . The order doesn't matter - both equal the same value, so they're the same base!

When do I subtract exponents versus add them?

Subtract when dividing: $\frac{a^m}{a^n} = a^{m-n}$
Add when multiplying: $a^m × a^n = a^{m+n}$
Think: division reduces the power!

What if the exponent in the denominator is larger?

You still subtract! If you get a negative exponent, that's correct. For example: $\frac{x^2}{x^5} = x^{2-5} = x^{-3}$

Do I need to calculate (7×2) first?

No! Keep it as $(7×2)$ in your answer. The question asks for the expression form, not the final numerical value.

How can I remember the subtraction rule?

Think of it this way: division cancels out multiplication. If you have $\frac{a×a×a}{a×a}$ , you cancel 2 a's from top and bottom, leaving one a. That's $a^{3-2} = a^1$ !

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