Question

Simplify each exponential expression using the properties of exponents and match it to the correct answer.

Answer 1

Answer:

• 2: B, D
• 1: A
• 1/2: C

Step-by-step explanation:

Here are the relevant rules of exponents:

  $a^{-b}=\dfrac{1}{a^b}\\\\a^0=1\\\\(a^b)(a^c)=a^{b+c}\\\\(a^b)^c=a^{bc}\\\\(ab)^c=(a^c)(b^c)$

Application

We can use 4 = 2² and 6 = 2·3 to expand each of the given expressions into powers of 2 and 3.

  $\boxed{\dfrac{3^{-3}\cdot2^{-3}\cdot6^3}{(4^0)^2}}=\dfrac{3^{-3}\cdot2^{-3}\cdot(2\cdot3)^3}{((2^2)^0)^2}=2^{-3+3-0}\cdot3^{-3+3}=1\cdot1=\boxed{1}$

  $\boxed{\dfrac{3^2\cdot4^3\cdot2^{-1}}{(3\cdot4)^2}}=\dfrac{3^2\cdot(2^2)^3\cdot2^{-1}}{3^2\cdot(2^2)^2}=2^{6-1-4}\cdot3^{2-2}=2\cdot1=\boxed{2}$

  $\boxed{\dfrac{2^4\cdot3^5}{(2\cdot3)^5}}=\dfrac{2^4\cdot3^5}{2^5\cdot3^5}=2^{4-5}\cdot3^{5-5}=2^{-1}\cdot1=\boxed{\dfrac{1}{2}}$

  $\boxed{\dfrac{(3\cdot2)^4\cdot3^{-3}}{2^3\cdot3}}=2^{4-3}\cdot3^{4-3-1}=2\cdot1=\boxed{2}$