Answer:
None of the expression are equivalent to [tex]49^{(2t - 0.5)}[/tex]
Step-by-step explanation:
Given
[tex]49^{(2t - 0.5)}[/tex]
Required
Find its equivalents
We start by expanding the given expression
[tex]49^{(2t - 0.5)}[/tex]
Expand 49
[tex](7^2)^{(2t - 0.5)}[/tex]
[tex]7^2^{(2t - 0.5)}[/tex]
Using laws of indices: [tex](a^m)^n = a^{mn}[/tex]
[tex]7^{(2*2t - 2*0.5)}[/tex]
[tex]7^{(4t - 1)}[/tex]
This implies that; each of the following options A,B and C must be equivalent to [tex]49^{(2t - 0.5)}[/tex] or alternatively, [tex]7^{(4t - 1)}[/tex]
A. [tex]\frac{7^{2t}}{49^{0.5}}[/tex]
Using law of indices which states;
[tex]a^{mn} = (a^m)^n[/tex]
Applying this law to the numerator; we have
[tex]\frac{(7^{2})^{t}}{49^{0.5}}[/tex]
Expand expression in bracket
[tex]\frac{(7 * 7)^{t}}{49^{0.5}}[/tex]
[tex]\frac{49^{t}}{49^{0.5}}[/tex]
Also; Using law of indices which states;
[tex]\frac{a^{m}}{a^n} = a^{m-n}[/tex]
[tex]\frac{49^{t}}{49^{0.5}}[/tex] becomes
[tex]49^{t-0.5}}[/tex]
This is not equivalent to [tex]49^{(2t - 0.5)}[/tex]
B. [tex]\frac{49^{2t}}{7^{0.5}}[/tex]
Expand numerator
[tex]\frac{(7*7)^{2t}}{7^{0.5}}[/tex]
[tex]\frac{(7^2)^{2t}}{7^{0.5}}[/tex]
Using law of indices which states;
[tex](a^m)^n = a^{mn}[/tex]
Applying this law to the numerator; we have
[tex]\frac{7^{2*2t}}{7^{0.5}}[/tex]
[tex]\frac{7^{4t}}{7^{0.5}}[/tex]
Also; Using law of indices which states;
[tex]\frac{a^{m}}{a^n} = a^{m-n}[/tex]
[tex]\frac{7^{4t}}{7^{0.5}}[/tex] = [tex]7^{4t - 0.5}[/tex]
This is also not equivalent to [tex]49^{(2t - 0.5)}[/tex]
C. [tex]7^{2t}\ *\ 49^{0.5}[/tex]
[tex]7^{2t}\ *\ (7^2)^{0.5}[/tex]
[tex]7^{2t}\ *\ 7^{2*0.5}[/tex]
[tex]7^{2t}\ *\ 7^{1}[/tex]
Using law of indices which states;
[tex]a^m*a^n = a^{m+n}[/tex]
[tex]7^{2t+ 1}[/tex]
This is also not equivalent to [tex]49^{(2t - 0.5)}[/tex]
