Question

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Answer 1

Question: Which equation is equivalent to $16^{2 p}=32^{p+3}?$

Option A: $8^{4 p}=8^{4 p+3}$

Option B: $8^{4 p}=8^{4 p+12}$

Option C: $2^{8 p}=2^{5 p+15}$

Option D: $2^{8 p}=2^{5 p+3}$

Answer:

Option C: $2^{8 p}=2^{5 p+15}$

Solution:

Given expression: $16^{2 p}=32^{p+3}$

Convert 16 to the base 2 and 32 to the base 2.

$(2^4)^{2 p}=(2^5)^{p+3}$

Using exponential rule: $\left(a^{b}\right)^{c}=a^{b c}$

$2^{8p}=2^{5p+15}$

Using the rule, $\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)$

8p = 5p + 15

3p = 15

p = 5

Substitute p = 5 in given expression, we get

$16^{10}=32^{8}\ \ \ \ \ \ \ \ \ \Rightarrow 1099511627776=1099511627776$

To find the equivalent expression to the given expression:

Option A: $8^{4 p}=8^{4 p+3}$

Using the rule, $\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)$

⇒ 4p = 4p + 3

⇒ 0p = 3

No solution for p, so it is not equivalent to the given expression.

Option B: $8^{4 p}=8^{4 p+12}$

Using the rule, $\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)$

⇒ 4p = 4p + 12

⇒ 0p = 12

No solution for p, so it is not equivalent to the given expression.

Option C: $2^{8 p}=2^{5 p+15}$

Using the rule, $\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)$

⇒ 8p = 5p + 15

⇒ 3p = 15

⇒ p = 5

Substitute p = 5 in $2^{8 p}=2^{5 p+15}$, we get

$2^{40}=2^{40}\ \ \ \ \ \ \ \ \ \Rightarrow 1099511627776=1099511627776$

It is equivalent to the given expression.

Option D: $2^{8 p}=2^{5 p+3}$

Using the rule, $\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)$

⇒ 8p = 5p + 3

⇒ 3p = 3

⇒ p = 1

Substitute p = 1 in $2^{8 p}=2^{5 p+3}$, we get

$2^{8}=2^{8}\ \ \ \ \ \ \ \ \ \Rightarrow 256=256$

It is not equivalent to the given expression.

Hence Option C is the correct answer.

Answer 2

Answer:

c

Step-by-step explanation: