Simplify (10×4)² ÷ (10×4)⁵: Power Division Problem

Q: Why do I subtract exponents when dividing?

Think of it as canceling out powers! When you divide $ \frac{a^5}{a^2} $, you're canceling 2 of the 5 factors, leaving 3. The rule $ \frac{a^m}{a^n} = a^{m-n} $ captures this pattern perfectly.

Q: What does a negative exponent mean?

A negative exponent means "flip it to the denominator"! So $ a^{-3} = \frac{1}{a^3} $. It's like the power is "upside down" from where you'd normally expect it.

Q: Do I need to calculate 10×4 first?

No! Keep $ (10×4) $ as one unit throughout the problem. The division rule works with any base, whether it's a number, expression, or variable.

Q: How can I check if my answer is correct?

Verify that $ \frac{1}{(10×4)^3} $ equals the original expression. You can also calculate: $ \frac{40^2}{40^5} = \frac{1600}{102400000} = \frac{1}{64000} = \frac{1}{40^3} $ ✓

Q: When do I get negative exponents?

You get negative exponents when the denominator has a larger exponent than the numerator. In $ \frac{a^2}{a^5} $, since 2 $ a^{2-5} = a^{-3} $.

Power Division with Negative Exponents

Insert the corresponding expression:

$\frac{\left(10\times4\right)^2}{\left(10\times4\right)^5}=$

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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 Let's use this formula in our exercise

00:14 Let's calculate the power

00:17 We'll use the formula for negative powers

00:20 Any number (A) to the power of (-N)

00:23 equals the reciprocal number (1/A) to the opposite power (N)

00:26 Let's use this formula in our exercise

00:28 We'll substitute the reciprocal number and the opposite power

00:30 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:

$\frac{\left(10\times4\right)^2}{\left(10\times4\right)^5}=$

Step-by-step solution

Let's solve the given expression step by step:
$\frac{\left(10\times4\right)^2}{\left(10\times4\right)^5}$

Step 1: Use the Power of a Quotient Rule for Exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$ . Here, $a = 10 \times 4$ , $m = 2$ , and $n = 5$

Therefore, apply the rule:

$\frac{\left(10\times4\right)^2}{\left(10\times4\right)^5} = \left(10\times4\right)^{2-5}$
$= \left(10\times4\right)^{-3}$

Step 2: Convert the expression with a negative exponent to a fraction:

We use the rule $a^{-n} = \frac{1}{a^n}$ .
Hence, $\left(10\times4\right)^{-3} = \frac{1}{\left(10\times4\right)^3}$ .

The solution to the question is: $\frac{1}{\left(10\times4\right)^3}$

Final Answer

$\frac{1}{\left(10\times4\right)^3}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: When dividing powers with same base: $\frac{a^m}{a^n} = a^{m-n}$
Technique: Subtract exponents: $(10×4)^{2-5} = (10×4)^{-3}$
Check: Convert negative exponent to fraction: $(10×4)^{-3} = \frac{1}{(10×4)^3}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add exponents like 2+5=7 when dividing powers! This gives $(10×4)^7$ which is completely wrong. Division means subtraction, so you get a negative exponent that becomes a fraction. Always subtract the bottom exponent from the top: 2-5=-3.

Practice Quiz

Test your knowledge with interactive questions

Insert the corresponding expression:

$\frac{9^{11}}{9^4}=$

FAQ

Everything you need to know about this question

Why do I subtract exponents when dividing?

Think of it as canceling out powers! When you divide $\frac{a^5}{a^2}$ , you're canceling 2 of the 5 factors, leaving 3. The rule $\frac{a^m}{a^n} = a^{m-n}$ captures this pattern perfectly.

What does a negative exponent mean?

A negative exponent means "flip it to the denominator"! So $a^{-3} = \frac{1}{a^3}$ . It's like the power is "upside down" from where you'd normally expect it.

Do I need to calculate 10×4 first?

No! Keep $(10×4)$ as one unit throughout the problem. The division rule works with any base, whether it's a number, expression, or variable.

How can I check if my answer is correct?

Verify that $\frac{1}{(10×4)^3}$ equals the original expression. You can also calculate: $\frac{40^2}{40^5} = \frac{1600}{102400000} = \frac{1}{64000} = \frac{1}{40^3}$ ✓

When do I get negative exponents?

You get negative exponents when the denominator has a larger exponent than the numerator. In $\frac{a^2}{a^5}$ , since 2 < 5, you get $a^{2-5} = a^{-3}$ .

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Simplify (10×4)² ÷ (10×4)⁵: Power Division Problem

Power Division with Negative Exponents

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I subtract exponents when dividing?

What does a negative exponent mean?

Do I need to calculate 10×4 first?

How can I check if my answer is correct?

When do I get negative exponents?

🌟 Unlock Your Math Potential

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