Question

Select all the expressions that are equivalent to (2)^n+³

A. (4)^n+²
B. 4(2)^2+¹
C. 8(2)^n
D. 16(2)^n
E. (2)^2x+³
Please explain as well please

Answer 1

The expressions which equivalent to
`(2)^(n+3)` are:

`4(2)^(n+1)` ⇒ B

`8(2)^(n)` ⇒ C

Explanation:

Let us revise some rules of exponent

• 
  `a^(m)` ×
  `a^(m)` =
  `a^(m+n)`
• 
  `(a^(m))^(n)` =
  `a^(m*n)`

Now let us find the equivalent expressions of
`(2)^(n+3)`

A.

∵ 4 = 2 × 2

∴ 4 =
`2^(2)`

∴
`(4)^(n+2)` =
`(2^(2))^(n+2)`

- By using the second rule above multiply 2 and (n + 2)

∵ 2(n + 2) = 2n + 4

∴
`(4)^(n+2)` =
`(2)^(2n+4)`

B.

∵ 4 = 2 × 2

∴ 4 = 2²

∴
`4(2)^(n+1)` = 2² ×
`(2)^(n+1)`

- By using the first rule rule add the exponents of 2

∵ 2 + n + 1 = n + 3

∴
`4(2)^(n+1)` =
`(2)^(n+3)`

C.

∵ 8 = 2 × 2 × 2

∴ 8 = 2³

∴
`8(2)^(n)` = 2³ ×
`(2)^(n)`

- By using the first rule rule add the exponents of 2

∵ 3 + n = n + 3

∴
`8(2)^(n)` =
`(2)^(n+3)`

D.

∵ 16 = 2 × 2 × 2 × 2

∴ 16 =
`2^(4)`

∴
`16(2)^(n)` =
`2^(4)` ×
`(2)^(n)`

- By using the first rule rule add the exponents of 2

∵ 4 + n = n + 4

∴
`16(2)^(n)` =
`(2)^(n+4)`

E.

`(2)^(2n+3)` is in its simplest form

The expressions which equivalent to
`(2)^(n+3)` are:

`4(2)^(n+1)` ⇒ B

`8(2)^(n)` ⇒ C