Question

Which are equivalent?

Answer 1

Answer:

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

Step-by-step 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) $\frac{50^x}{5}$

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

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

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

$=\frac{2^x\cdot \:5^{2x}}{5}$

Apply exponent rule , $\frac{x^a}{x^b}\:=\:x^{a-b}$

$=2^x\cdot \:5^{2x-1}$

$\frac{50^x}{5}$ is not equivalent to given expression $10^x$

c) $x^5$

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

d) $\:\left(\frac{50}{5}\:\right)^x$

Divide 50 by 5 we have 10

So $\:\left(\frac{50}{5}\:\right)^x=10^x$

$\:\left(\frac{50}{5}\:\right)^x$ is equivalent to given expression $10^x$

e) $\frac{50^x}{5^x}$

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

$=\frac{2^x\cdot \:5^{2x}}{5^x}$

Apply exponent rule , $\frac{x^a}{x^b}\:=\:x^{a-b}$

$=2^x\cdot \:5^{2x-x}$

$=2^x\cdot \:5^{x}$

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

$=2^x\cdot \:5^{x}=10^x$

$\frac{50^x}{5^x}$ 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.

Answer 2

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