Question

Which are equivalent?

Answer 1

d and e are equivalent
if you plug in 1,2,3 for X for each equation you can see only d and e are equivalent to the original this is because (50/5)^x and 50^x/5^x simplified is still 10^x

Answer 2

Option (a), (d) and (e) are correct.

Explanation:

Given : expression
`10^x`

We have to select the equivalent fractions from the given options.

We will check each given option one by one,

a)
`10\cdot 10^(x-1)`

Apply property of exponents,
`a^b\cdot \:a^c=a^(b+c)`

We have,
`10\cdot \:10^(x-1)=\:10^(1+x-1)=\:10^x`

`10\cdot 10^(x-1)` is equivalent to given expression
`10^x`

b)
`(50^x)/(5)`

Breaking 50 into factor as
`50=5^2\cdot \:2`

Thus,
`=\left(5^2\cdot \:2\right)^x`

Apply exponent rule ,
`\left(ab\right)^c=a^cb^c`

`=(2^x\cdot \:5^(2x))/(5)`

Apply exponent rule ,
`(x^a)/(x^b)\:=\:x^(a-b)`

`=2^x\cdot \:5^(2x-1)`

`(50^x)/(5)` is not equivalent to given expression
`10^x`

c)
`x^5`

Clearly seen
`x^5` is not equivalent to given expression
`10^x`

d)
`\:\left((50)/(5)\:\right)^x`

Divide 50 by 5 we have 10

So
`\:\left((50)/(5)\:\right)^x=10^x`

`\:\left((50)/(5)\:\right)^x` is equivalent to given expression
`10^x`

e)
`(50^x)/(5^x)`

Breaking 50 into factor as
`50=5^2\cdot \:2`

`=(2^x\cdot \:5^(2x))/(5^x)`

Apply exponent rule ,
`(x^a)/(x^b)\:=\:x^(a-b)`

`=2^x\cdot \:5^(2x-x)`

`=2^x\cdot \:5^(x)`

Apply exponent rule
`a^mb^m=\left(ab\right)^m`

`=2^x\cdot \:5^(x)=10^x`

`(50^x)/(5^x)` is equivalent to given expression
`10^x`

f)
`10\cdot 10^(x+1)`

Apply property of exponents,
`a^b\cdot \:a^c=a^(b+c)`

We have,
`10\cdot \:10^(x+1)=\:10^(1+x+1)=\:10^(x+2)`

`10\cdot 10^(x+2)` is not equivalent to given expression
`10^x`

Thus, option (a), (d) and (e) are correct.