Get detailed and reliable answers to your questions on IDNLearn.com. Get the information you need from our experts, who provide reliable and detailed answers to all your questions.
Sagot :
Sure! Let's solve the given equation step by step. We need to solve the equation:
[tex]\[ \left(\sqrt{\frac{3}{5}}\right)^{x-1} = \left(\frac{27}{125}\right)^{-1} \][/tex]
### Step 1: Simplify the right side of the equation
First, we can simplify the right-hand side of the equation. Recall that:
[tex]\[ \left(\frac{a}{b}\right)^{-1} = \frac{b}{a} \][/tex]
So:
[tex]\[ \left(\frac{27}{125}\right)^{-1} = \frac{125}{27} \][/tex]
### Step 2: Substitute and rewrite the equation
Now the equation becomes:
[tex]\[ \left(\sqrt{\frac{3}{5}}\right)^{x-1} = \frac{125}{27} \][/tex]
### Step 3: Simplify the left side of the equation
Next, look at the left-hand side of the equation. Recall that:
[tex]\[ \sqrt{\frac{3}{5}} = \left(\frac{3}{5}\right)^{1/2} \][/tex]
So we rewrite the left-hand side as:
[tex]\[ \left(\left(\frac{3}{5}\right)^{1/2}\right)^{x-1} \][/tex]
Using the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[ \left(\frac{3}{5}\right)^{(1/2)(x-1)} = \left(\frac{3}{5}\right)^{\frac{x-1}{2}} \][/tex]
### Step 4: Equate and solve exponents
We now have:
[tex]\[ \left(\frac{3}{5}\right)^{\frac{x-1}{2}} = \frac{125}{27} \][/tex]
Notice that [tex]\(\frac{125}{27}\)[/tex] can be written as [tex]\(\left(\frac{5}{3}\right)^3\)[/tex]:
[tex]\[ \frac{125}{27} = \left(\frac{5}{3}\right)^3 \][/tex]
We then rewrite the equation as:
[tex]\[ \left(\frac{3}{5}\right)^{\frac{x-1}{2}} = \left(\frac{5}{3}\right)^3 \][/tex]
### Step 5: Align bases and solve for x
Since [tex]\(\frac{3}{5}\)[/tex] is the reciprocal of [tex]\(\frac{5}{3}\)[/tex], we can rewrite [tex]\(\left(\frac{5}{3}\right)^3\)[/tex] as [tex]\(\left(\frac{3}{5}\right)^{-3}\)[/tex]:
[tex]\[ \left(\frac{3}{5}\right)^{\frac{x-1}{2}} = \left(\frac{3}{5}\right)^{-3} \][/tex]
If the bases are equal, we can set their exponents equal to each other:
[tex]\[ \frac{x-1}{2} = -3 \][/tex]
### Step 6: Solve for x
Multiply both sides by 2 to clear the fraction:
[tex]\[ x - 1 = -6 \][/tex]
Finally, add 1 to both sides to isolate x:
[tex]\[ x = -6 + 1 \][/tex]
[tex]\[ x = -5 \][/tex]
So, the solution to the equation is [tex]\( x = -5 \)[/tex].
[tex]\[ \left(\sqrt{\frac{3}{5}}\right)^{x-1} = \left(\frac{27}{125}\right)^{-1} \][/tex]
### Step 1: Simplify the right side of the equation
First, we can simplify the right-hand side of the equation. Recall that:
[tex]\[ \left(\frac{a}{b}\right)^{-1} = \frac{b}{a} \][/tex]
So:
[tex]\[ \left(\frac{27}{125}\right)^{-1} = \frac{125}{27} \][/tex]
### Step 2: Substitute and rewrite the equation
Now the equation becomes:
[tex]\[ \left(\sqrt{\frac{3}{5}}\right)^{x-1} = \frac{125}{27} \][/tex]
### Step 3: Simplify the left side of the equation
Next, look at the left-hand side of the equation. Recall that:
[tex]\[ \sqrt{\frac{3}{5}} = \left(\frac{3}{5}\right)^{1/2} \][/tex]
So we rewrite the left-hand side as:
[tex]\[ \left(\left(\frac{3}{5}\right)^{1/2}\right)^{x-1} \][/tex]
Using the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[ \left(\frac{3}{5}\right)^{(1/2)(x-1)} = \left(\frac{3}{5}\right)^{\frac{x-1}{2}} \][/tex]
### Step 4: Equate and solve exponents
We now have:
[tex]\[ \left(\frac{3}{5}\right)^{\frac{x-1}{2}} = \frac{125}{27} \][/tex]
Notice that [tex]\(\frac{125}{27}\)[/tex] can be written as [tex]\(\left(\frac{5}{3}\right)^3\)[/tex]:
[tex]\[ \frac{125}{27} = \left(\frac{5}{3}\right)^3 \][/tex]
We then rewrite the equation as:
[tex]\[ \left(\frac{3}{5}\right)^{\frac{x-1}{2}} = \left(\frac{5}{3}\right)^3 \][/tex]
### Step 5: Align bases and solve for x
Since [tex]\(\frac{3}{5}\)[/tex] is the reciprocal of [tex]\(\frac{5}{3}\)[/tex], we can rewrite [tex]\(\left(\frac{5}{3}\right)^3\)[/tex] as [tex]\(\left(\frac{3}{5}\right)^{-3}\)[/tex]:
[tex]\[ \left(\frac{3}{5}\right)^{\frac{x-1}{2}} = \left(\frac{3}{5}\right)^{-3} \][/tex]
If the bases are equal, we can set their exponents equal to each other:
[tex]\[ \frac{x-1}{2} = -3 \][/tex]
### Step 6: Solve for x
Multiply both sides by 2 to clear the fraction:
[tex]\[ x - 1 = -6 \][/tex]
Finally, add 1 to both sides to isolate x:
[tex]\[ x = -6 + 1 \][/tex]
[tex]\[ x = -5 \][/tex]
So, the solution to the equation is [tex]\( x = -5 \)[/tex].
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.