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

Question

Insert the corresponding expression:

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

Video Solution

Solution Steps

00:00 Simply
00:06 In multiplication, the order of factors doesn't matter
00:10 We'll use this formula in our exercise and switch between the factors
00:19 According to the laws of exponents, division of exponents with equal bases (A)
00:23 equals the same base (A) raised to the power of the difference of exponents (M-N)
00:26 We'll use this formula in our exercise
00:29 And this is the solution to the question

Step-by-Step Solution

Let's simplify the expression (15×4)3(4×15)9 \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: (15×4) (15 \times 4) . Therefore, we have a quotient of powers with the same base:

(15×4)3(15×4)9 \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:

(15×4)39 (15 \times 4)^{3 - 9}

Subtracting the exponents, we have:

(15×4)6 (15 \times 4)^{-6}

This matches with one of the choices:

  • Choice 1: (15×4)39(15\times4)^{3-9} is correct, as it simplifies to (15×4)6 (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:
(15×4)39 \left(15\times4\right)^{3-9} .

Answer

(15×4)39 \left(15\times4\right)^{3-9}