Third choice, option C
x = 5/2
To solve for the value of x if 9^(x-1) - 2 = 25, we can substitute each of the options into a simplified equation.
We could substitute each option into the given equation, but then we would have to simplify it four more times. It's easier to simplify once first.
[tex]9^{x-1} -2=25[/tex] Start with the given equation
[tex]9^{x-1} -2+2=25+2[/tex] Add 2 to both sides
[tex]9^{x-1} =25+2[/tex] –2 + 2 on the left side cancels out
[tex]9^{x-1} =27[/tex] Add on the right side
We found a simplified equation.
Now, let's substitute each option choice for 'x' in the simplified equation. We only have to substitute choices until we find the choice that makes the left and right sides of the equation equal, but let's try all of the choices today.
[tex]9^{x-1} =27[/tex] Start with the simplified equation
[tex]\displaystyle{9^{\frac{1}{2}-1} =27}[/tex] Substitute x = 1/2
[tex]\displaystyle{9^{\frac{1}{2}-\frac{2}{2}} =27}[/tex] Find a common denominator within the exponent
[tex]\displaystyle{9^{\frac{1-2}{2}} =27}[/tex] Combine numerators and subtract
[tex]\displaystyle{9^{-\frac{1}{2}} =27}[/tex]
[tex]\displaystyle{(3^{2})^{-\frac{1}{2}} =27}[/tex] Convert 9 into a base and exponent
[tex]\displaystyle{ 3^{ 2*-\frac{1}{2} } =27 }[/tex] Multiply exponents
[tex]\displaystyle{ 3^{ -\frac{2*1}{2} } =27 }[/tex] Combine into numerator
[tex]\displaystyle{ 3^{ -\frac{2}{2} } =27 }[/tex] Simplify the fraction
[tex]\displaystyle{ 3^{ -1 } =27 }[/tex] Use negative exponent rule to simplify
[tex]\displaystyle{ \frac{1}{3^{1}} =27 }[/tex] Simplify the denominator
[tex]\displaystyle{ \frac{1}{3} \neq 27 }[/tex] Left and right sides are not equal
So, x is not equal to 1/2.
[tex]9^{x-1} =27[/tex] Start with the simplified equation
[tex]9^{2-1} =27[/tex] Substitute x = 2
[tex]9^{1} =27[/tex] Subtract within the exponent
[tex]9 \neq 27[/tex] Left and right sides are not equal
So, x is not equal to 2.
[tex]9^{x-1} =27[/tex] Start with the simplified equation
[tex]\displaystyle{9^{\frac{5}{2}-1} =27}[/tex] Substitute x = 5/2
[tex]\displaystyle{9^{\frac{5}{2}-\frac{2}{2}} =27}[/tex] Find a common denominator within the exponent
[tex]\displaystyle{9^{\frac{5-2}{2}} =27}[/tex] Combine numerators and subtract
[tex]\displaystyle{9^{\frac{3}{2}} =27}[/tex] Use fractional exponent rule to simplify
[tex]\displaystyle{\sqrt[2]{9^{3}} =27}[/tex]
[tex]\displaystyle{\sqrt{9^{3}} =27}[/tex]
[tex]\displaystyle{\sqrt{9 * 9 * 9} =27}[/tex] Solve the exponent by multiplying bases
[tex]\displaystyle{\sqrt{729} =27}[/tex] Solve the root
[tex]\displaystyle{27 =27}[/tex] Left and right sides are equal
So, x is equal to 5/2.
[tex]9^{x-1} =27[/tex] Start with the simplified equation
[tex]9^{4-1} =27[/tex] Substitute x = 4
[tex]9^{3} =27[/tex] Subtract within the exponent
[tex]9*9*9 =27[/tex] Solve the exponent by multiplying bases
[tex]729 \neq 27[/tex] Left and right sides are not equal
So, x is not equal to 4.
∴ in [tex]9^{x-1} -2=25[/tex], x is equal to [tex]\displatestyle{\frac{5}{2}}[/tex], making option C correct.
Learn more about exponent rules here:
https://brainly.com/question/84608
https://brainly.com/question/12173411
