Simplify (15×4)³/(15×4)⁹: Working with Power Laws

Q: Why can I treat (15×4) and (4×15) as the same base?

Because multiplication is commutative, meaning the order doesn't matter! $ 15×4 = 4×15 = 60 $, so they represent the exact same base value.

Q: What does the negative exponent -6 actually mean?

A negative exponent means reciprocal! So $ (15×4)^{-6} = \frac{1}{(15×4)^6} $. The expression becomes a fraction with 1 on top.

Q: Do I need to calculate 15×4=60 first?

Not necessary! You can work with $ (15×4) $ as a single base throughout the problem. Only calculate the actual number if the question asks for a numerical answer.

Q: How do I remember when to subtract exponents?

Think "Division = Subtraction" for exponents! When you see a fraction with the same base, always subtract: top exponent minus bottom exponent.

Q: What if the exponents were in different order, like 9-3?

The order matters! $ \frac{a^9}{a^3} = a^{9-3} = a^6 $ is different from $ \frac{a^3}{a^9} = a^{3-9} = a^{-6} $. Always subtract denominator from numerator.

Quotient Rule with Same Base Expressions

Insert the corresponding expression:

$\frac{\left(15\times4\right)^3}{\left(4\times15\right)^9}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 In multiplication, the order of factors doesn't matter

00:09 We'll use this formula in our exercise and reverse the order of factors

00:18 We'll use the formula for dividing powers

00:21 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:24 equals the number (A) to the power of the difference of exponents (M-N)

00:27 We'll use this formula in our exercise

00:29 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:

$\frac{\left(15\times4\right)^3}{\left(4\times15\right)^9}=$

Step-by-step solution

Let's simplify the expression $\frac{\left(15\times4\right)^3}{\left(4\times15\right)^9}$ :

We start by recognizing that both the numerator and the denominator share the same base: $(15 \times 4)$ . Therefore, we have a quotient of powers with the same base:

$\frac{\left(15\times4\right)^3}{\left(15\times4\right)^9}$

According to the rules of exponents, when dividing like bases, we subtract the exponents:

$(15 \times 4)^{3 - 9}$

Subtracting the exponents, we have:

$(15 \times 4)^{-6}$

This matches with one of the choices:

Choice 1: $(15\times4)^{3-9}$ is correct, as it simplifies to $(15\times4)^{-6}$ .
Choice 2 and Choice 3 involve incorrect operations on exponents (multiplication and addition respectively).
Choice 4 reverses the subtraction order, which results differently in base powers than what is needed.

Therefore, the correct answer to the problem is:
$\left(15\times4\right)^{3-9}$ .

Final Answer

$\left(15\times4\right)^{3-9}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract the exponents
Technique: $\frac{a^m}{a^n} = a^{m-n}$ so $\frac{(15×4)^3}{(15×4)^9} = (15×4)^{3-9}$
Check: Verify the base is identical: (15×4) = (4×15) by commutative property ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents like 3+9=12 or multiply like 3×9=27 when dividing powers! This gives completely wrong results like $(15×4)^{12}$ instead of $(15×4)^{-6}$ . Always subtract the bottom exponent from the top exponent when dividing same bases.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I treat (15×4) and (4×15) as the same base?

Because multiplication is commutative, meaning the order doesn't matter! $15×4 = 4×15 = 60$ , so they represent the exact same base value.

What does the negative exponent -6 actually mean?

A negative exponent means reciprocal! So $(15×4)^{-6} = \frac{1}{(15×4)^6}$ . The expression becomes a fraction with 1 on top.

Do I need to calculate 15×4=60 first?

Not necessary! You can work with $(15×4)$ as a single base throughout the problem. Only calculate the actual number if the question asks for a numerical answer.

How do I remember when to subtract exponents?

Think "Division = Subtraction" for exponents! When you see a fraction with the same base, always subtract: top exponent minus bottom exponent.

What if the exponents were in different order, like 9-3?

The order matters! $\frac{a^9}{a^3} = a^{9-3} = a^6$ is different from $\frac{a^3}{a^9} = a^{3-9} = a^{-6}$ . Always subtract denominator from numerator.

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