Question: Which equation is equivalent to ![16^{2 p}=32^{p+3}?](https://tex.z-dn.net/?f=16%5E%7B2%20p%7D%3D32%5E%7Bp%2B3%7D%3F)
Option A: ![8^{4 p}=8^{4 p+3}](https://tex.z-dn.net/?f=8%5E%7B4%20p%7D%3D8%5E%7B4%20p%2B3%7D)
Option B: ![8^{4 p}=8^{4 p+12}](https://tex.z-dn.net/?f=8%5E%7B4%20p%7D%3D8%5E%7B4%20p%2B12%7D)
Option C: ![2^{8 p}=2^{5 p+15}](https://tex.z-dn.net/?f=2%5E%7B8%20p%7D%3D2%5E%7B5%20p%2B15%7D)
Option D: ![2^{8 p}=2^{5 p+3}](https://tex.z-dn.net/?f=2%5E%7B8%20p%7D%3D2%5E%7B5%20p%2B3%7D)
Answer:
Option C: ![2^{8 p}=2^{5 p+15}](https://tex.z-dn.net/?f=2%5E%7B8%20p%7D%3D2%5E%7B5%20p%2B15%7D)
Solution:
Given expression: ![16^{2 p}=32^{p+3}](https://tex.z-dn.net/?f=16%5E%7B2%20p%7D%3D32%5E%7Bp%2B3%7D)
Convert 16 to the base 2 and 32 to the base 2.
![(2^4)^{2 p}=(2^5)^{p+3}](https://tex.z-dn.net/?f=%282%5E4%29%5E%7B2%20p%7D%3D%282%5E5%29%5E%7Bp%2B3%7D)
Using exponential rule: ![\left(a^{b}\right)^{c}=a^{b c}](https://tex.z-dn.net/?f=%5Cleft%28a%5E%7Bb%7D%5Cright%29%5E%7Bc%7D%3Da%5E%7Bb%20c%7D)
![2^{8p}=2^{5p+15}](https://tex.z-dn.net/?f=2%5E%7B8p%7D%3D2%5E%7B5p%2B15%7D)
Using the rule, ![\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)](https://tex.z-dn.net/?f=%5Ctext%20%7B%20If%20%7D%20d%5E%7Bf%28x%29%7D%3Dd%5E%7Bg%28x%29%7D%2C%20%5Ctext%20%7B%20then%20%7D%20f%28x%29%3Dg%28x%29)
8p = 5p + 15
3p = 15
p = 5
Substitute p = 5 in given expression, we get
![16^{10}=32^{8}\ \ \ \ \ \ \ \ \ \Rightarrow 1099511627776=1099511627776](https://tex.z-dn.net/?f=16%5E%7B10%7D%3D32%5E%7B8%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%20%5CRightarrow%201099511627776%3D1099511627776)
<u>To find the equivalent expression to the given expression:</u>
Option A: ![8^{4 p}=8^{4 p+3}](https://tex.z-dn.net/?f=8%5E%7B4%20p%7D%3D8%5E%7B4%20p%2B3%7D)
Using the rule, ![\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)](https://tex.z-dn.net/?f=%5Ctext%20%7B%20If%20%7D%20d%5E%7Bf%28x%29%7D%3Dd%5E%7Bg%28x%29%7D%2C%20%5Ctext%20%7B%20then%20%7D%20f%28x%29%3Dg%28x%29)
⇒ 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}](https://tex.z-dn.net/?f=8%5E%7B4%20p%7D%3D8%5E%7B4%20p%2B12%7D)
Using the rule, ![\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)](https://tex.z-dn.net/?f=%5Ctext%20%7B%20If%20%7D%20d%5E%7Bf%28x%29%7D%3Dd%5E%7Bg%28x%29%7D%2C%20%5Ctext%20%7B%20then%20%7D%20f%28x%29%3Dg%28x%29)
⇒ 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}](https://tex.z-dn.net/?f=2%5E%7B8%20p%7D%3D2%5E%7B5%20p%2B15%7D)
Using the rule, ![\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)](https://tex.z-dn.net/?f=%5Ctext%20%7B%20If%20%7D%20d%5E%7Bf%28x%29%7D%3Dd%5E%7Bg%28x%29%7D%2C%20%5Ctext%20%7B%20then%20%7D%20f%28x%29%3Dg%28x%29)
⇒ 8p = 5p + 15
⇒ 3p = 15
⇒ p = 5
Substitute p = 5 in
, we get
![2^{40}=2^{40}\ \ \ \ \ \ \ \ \ \Rightarrow 1099511627776=1099511627776](https://tex.z-dn.net/?f=2%5E%7B40%7D%3D2%5E%7B40%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%20%5CRightarrow%201099511627776%3D1099511627776)
It is equivalent to the given expression.
Option D: ![2^{8 p}=2^{5 p+3}](https://tex.z-dn.net/?f=2%5E%7B8%20p%7D%3D2%5E%7B5%20p%2B3%7D)
Using the rule, ![\text { If } d^{f(x)}=d^{g(x)}, \text { then } f(x)=g(x)](https://tex.z-dn.net/?f=%5Ctext%20%7B%20If%20%7D%20d%5E%7Bf%28x%29%7D%3Dd%5E%7Bg%28x%29%7D%2C%20%5Ctext%20%7B%20then%20%7D%20f%28x%29%3Dg%28x%29)
⇒ 8p = 5p + 3
⇒ 3p = 3
⇒ p = 1
Substitute p = 1 in
, we get
![2^{8}=2^{8}\ \ \ \ \ \ \ \ \ \Rightarrow 256=256](https://tex.z-dn.net/?f=2%5E%7B8%7D%3D2%5E%7B8%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%5C%20%20%5CRightarrow%20256%3D256)
It is not equivalent to the given expression.
Hence Option C is the correct answer.
Answer:
27
Step-by-step explanation:
The formula for the volume of a cube is given as
. if s=3in, then V=27inches cubed.
Answer:
The answer is 2
Step-by-step explanation:
When a line is parallel to the other line, they must have the same gradient.
Answer:
<h2>C. y = 2x + 5</h2>
Step-by-step explanation:
The slope-intercept form of an equation of a line:
<em>y = mx + b</em>
<em>m</em><em> - slope</em>
<em>b</em><em> - y-intercept</em>
<em />
We have
<em>m = 2, b = 5</em>
Substitute:
y = 2x + 5
Answer:
(a+2) Is your answer
Step-by-step explanation:
You could try synthetic division, but it's pretty complicated (not to mention for much harder problems), so you should try factoring this. What two numbers multiply up to 10 and add up to 7? Using trial and error, we would find these two numbers to be 2 and 5. Therefore, it factors into (a+5)(a+2).