Discover new information and insights with the help of IDNLearn.com. Ask your questions and receive reliable and comprehensive answers from our dedicated community of professionals.

If [tex]3^{2x-1} = 9^{\frac{-x}{2}}[/tex], then [tex]x[/tex] is equal to:

Sagot :

To solve the equation [tex]\(3^{2x - 1} = 9^{\frac{-x}{2}}\)[/tex], let’s break it down step by step:

1. Rewrite the bases to be the same:
The base of the exponent on the right side, 9, can be rewritten in terms of 3. Since [tex]\(9 = 3^2\)[/tex], the equation becomes:
[tex]\[ 9^{\frac{-x}{2}} = (3^2)^{\frac{-x}{2}} \][/tex]

2. Simplify the exponent:
Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we can simplify the right-hand side:
[tex]\[ (3^2)^{\frac{-x}{2}} = 3^{2 \cdot \frac{-x}{2}} = 3^{-x} \][/tex]
Now the equation is:
[tex]\[ 3^{2x - 1} = 3^{-x} \][/tex]

3. Set the exponents equal to each other:
Since the bases are now the same, we can set the exponents equal to each other:
[tex]\[ 2x - 1 = -x \][/tex]

4. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], combine like terms (i.e., get all [tex]\( x \)[/tex]-terms on one side of the equation):
[tex]\[ 2x + x - 1 = 0 \][/tex]
Simplify:
[tex]\[ 3x - 1 = 0 \][/tex]
Add 1 to both sides:
[tex]\[ 3x = 1 \][/tex]
Divide both sides by 3:
[tex]\[ x = \frac{1}{3} \][/tex]

Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[ x = \frac{1}{3} \][/tex]

Converting [tex]\(\frac{1}{3}\)[/tex] to a decimal, we get approximately [tex]\( 0.3333333333333333 \)[/tex].
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Your search for solutions ends at IDNLearn.com. Thank you for visiting, and we look forward to helping you again.