Answer:
Option (a), (d) and (e) are correct.
Step-by-step explanation:
Given : expression [tex]10^x[/tex]
We have to select the equivalent fractions from the given options.
We will check each given option one by one,
a) [tex]10\cdot 10^{x-1}[/tex]
Apply property of exponents, [tex]a^b\cdot \:a^c=a^{b+c}[/tex]
We have, [tex]10\cdot \:10^{x-1}=\:10^{1+x-1}=\:10^x[/tex]
[tex]10\cdot 10^{x-1}[/tex] is equivalent to given expression [tex]10^x[/tex]
b) [tex]\frac{50^x}{5}[/tex]
Breaking 50 into factor as [tex]50=5^2\cdot \:2[/tex]
Thus, [tex]=\left(5^2\cdot \:2\right)^x[/tex]
Apply exponent rule , [tex]\left(ab\right)^c=a^cb^c[/tex]
[tex]=\frac{2^x\cdot \:5^{2x}}{5}[/tex]
Apply exponent rule , [tex]\frac{x^a}{x^b}\:=\:x^{a-b}[/tex]
[tex]=2^x\cdot \:5^{2x-1}[/tex]
[tex]\frac{50^x}{5}[/tex] is not equivalent to given expression [tex]10^x[/tex]
c) [tex]x^5[/tex]
Clearly seen [tex]x^5[/tex] is not equivalent to given expression [tex]10^x[/tex]
d) [tex]\:\left(\frac{50}{5}\:\right)^x[/tex]
Divide 50 by 5 we have 10
So [tex]\:\left(\frac{50}{5}\:\right)^x=10^x[/tex]
[tex]\:\left(\frac{50}{5}\:\right)^x[/tex] is equivalent to given expression [tex]10^x[/tex]
e) [tex]\frac{50^x}{5^x}[/tex]
Breaking 50 into factor as [tex]50=5^2\cdot \:2[/tex]
[tex]=\frac{2^x\cdot \:5^{2x}}{5^x}[/tex]
Apply exponent rule , [tex]\frac{x^a}{x^b}\:=\:x^{a-b}[/tex]
[tex]=2^x\cdot \:5^{2x-x}[/tex]
[tex]=2^x\cdot \:5^{x}[/tex]
Apply exponent rule [tex]a^mb^m=\left(ab\right)^m[/tex]
[tex]=2^x\cdot \:5^{x}=10^x[/tex]
[tex]\frac{50^x}{5^x}[/tex] is equivalent to given expression [tex]10^x[/tex]
f) [tex]10\cdot 10^{x+1}[/tex]
Apply property of exponents, [tex]a^b\cdot \:a^c=a^{b+c}[/tex]
We have, [tex]10\cdot \:10^{x+1}=\:10^{1+x+1}=\:10^{x+2}[/tex]
[tex]10\cdot 10^{x+2}[/tex] is not equivalent to given expression [tex]10^x[/tex]
Thus, option (a), (d) and (e) are correct.
