Respuesta :
Question: Which equation is equivalent to [tex]16^{2 p}=32^{p+3}?[/tex]
Option A: [tex]8^{4 p}=8^{4 p+3}[/tex]
Option B: [tex]8^{4 p}=8^{4 p+12}[/tex]
Option C: [tex]2^{8 p}=2^{5 p+15}[/tex]
Option D: [tex]2^{8 p}=2^{5 p+3}[/tex]
Answer:
Option C: [tex]2^{8 p}=2^{5 p+15}[/tex]
Solution:
Given expression: [tex]16^{2 p}=32^{p+3}[/tex]
Convert 16 to the base 2 and 32 to the base 2.
[tex](2^4)^{2 p}=(2^5)^{p+3}[/tex]
Using exponential rule: [tex]\left(a^{b}\right)^{c}=a^{b c}[/tex]
[tex]2^{8p}=2^{5p+15}[/tex]
Using the rule, [tex]\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)[/tex]
8p = 5p + 15
3p = 15
p = 5
Substitute p = 5 in given expression, we get
[tex]16^{10}=32^{8}\ \ \ \ \ \ \ \ \ \Rightarrow 1099511627776=1099511627776[/tex]
To find the equivalent expression to the given expression:
Option A: [tex]8^{4 p}=8^{4 p+3}[/tex]
Using the rule, [tex]\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)[/tex]
⇒ 4p = 4p + 3
⇒ 0p = 3
No solution for p, so it is not equivalent to the given expression.
Option B: [tex]8^{4 p}=8^{4 p+12}[/tex]
Using the rule, [tex]\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)[/tex]
⇒ 4p = 4p + 12
⇒ 0p = 12
No solution for p, so it is not equivalent to the given expression.
Option C: [tex]2^{8 p}=2^{5 p+15}[/tex]
Using the rule, [tex]\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)[/tex]
⇒ 8p = 5p + 15
⇒ 3p = 15
⇒ p = 5
Substitute p = 5 in [tex]2^{8 p}=2^{5 p+15}[/tex], we get
[tex]2^{40}=2^{40}\ \ \ \ \ \ \ \ \ \Rightarrow 1099511627776=1099511627776[/tex]
It is equivalent to the given expression.
Option D: [tex]2^{8 p}=2^{5 p+3}[/tex]
Using the rule, [tex]\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)[/tex]
⇒ 8p = 5p + 3
⇒ 3p = 3
⇒ p = 1
Substitute p = 1 in [tex]2^{8 p}=2^{5 p+3}[/tex], we get
[tex]2^{8}=2^{8}\ \ \ \ \ \ \ \ \ \Rightarrow 256=256[/tex]
It is not equivalent to the given expression.
Hence Option C is the correct answer.
Answer:
Which equation is equivalent to [tex]16^{2p} = 32^{p+3}[/tex] ?
a.) [tex]8^{4p} = 8^{4p+3}[/tex]
b.) [tex]8^{4p} = 8^{4p+12}[/tex]
c.) [tex]2^{8p} = 2^{5p+15}[/tex] (Answer)
d.) [tex]2^{8p} = 2^{5p+3}[/tex]
Step-by-step explanation:
Edge 2021-2022 :)
