Simplify (4×7)^12 ÷ (4×7)^5: Exponent Division Problem

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{a^{12}}{a^5} $ means you have 12 copies of a in the numerator and 5 copies in the denominator. When you cancel out 5 from both top and bottom, you're left with 12 - 5 = 7 copies of a.

Q: Do I need to calculate what 4×7 equals first?

No! Keep the base as $ (4×7) $ throughout the problem. The quotient rule works with any identical base, whether it's a number, variable, or expression like $ (4×7) $.

Q: What if the bottom exponent is bigger than the top?

You still subtract! For example, $ \frac{x^3}{x^8} = x^{3-8} = x^{-5} $. The negative exponent means one over that positive power: $ x^{-5} = \frac{1}{x^5} $.

Q: How can I remember the quotient rule?

Think: "Same base, subtract!" When you see identical bases being divided, you always subtract exponents. It's the opposite of multiplication where you add exponents.

Q: Can I use this rule with different bases?

No! The quotient rule only works when the bases are exactly the same. For example, $ \frac{2^5}{3^2} $ cannot be simplified using this rule because 2 and 3 are different bases.

Exponent Division with Like Bases

Insert the corresponding expression:

$\frac{\left(4\times7\right)^{12}}{\left(4\times7\right)^5}=$

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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 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 Let's calculate the power

00:14 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:

$\frac{\left(4\times7\right)^{12}}{\left(4\times7\right)^5}=$

Step-by-step solution

The given expression is $\frac{\left(4\times7\right)^{12}}{\left(4\times7\right)^5}$ .
We are asked to simplify this expression using the Power of a Quotient Rule for Exponents.

The Power of a Quotient Rule states:

When you divide like bases, you subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$ .

Applying this rule to the given problem:

1. The base of both the numerator and the denominator is $4 \times 7$ .
2. The exponent in the numerator is 12, and the exponent in the denominator is 5.
3. Therefore, subtract the exponents: $12 - 5 = 7$ .

The simplified expression becomes:

$\left(4\times7\right)^7$ .

The solution to the question is: $\left(4\times7\right)^7$

Final Answer

$\left(4\times7\right)^7$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing like bases, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: Identify base $(4×7)$ appears in both numerator and denominator
Check: Verify $12 - 5 = 7$ gives $(4×7)^7$ ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents (12 + 5 = 17) or multiply them (12 × 5 = 60) when dividing! This completely ignores the division operation. Always subtract the bottom exponent from the top exponent when bases are identical.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

Think of it this way: $\frac{a^{12}}{a^5}$ means you have 12 copies of a in the numerator and 5 copies in the denominator. When you cancel out 5 from both top and bottom, you're left with 12 - 5 = 7 copies of a.

Do I need to calculate what 4×7 equals first?

No! Keep the base as $(4×7)$ throughout the problem. The quotient rule works with any identical base, whether it's a number, variable, or expression like $(4×7)$ .

What if the bottom exponent is bigger than the top?

You still subtract! For example, $\frac{x^3}{x^8} = x^{3-8} = x^{-5}$ . The negative exponent means one over that positive power: $x^{-5} = \frac{1}{x^5}$ .

How can I remember the quotient rule?

Think: "Same base, subtract!" When you see identical bases being divided, you always subtract exponents. It's the opposite of multiplication where you add exponents.

Can I use this rule with different bases?

No! The quotient rule only works when the bases are exactly the same. For example, $\frac{2^5}{3^2}$ cannot be simplified using this rule because 2 and 3 are different bases.

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