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Sagot :
To determine the value of [tex]\( x \)[/tex] in the equation [tex]\(\left(\frac{1}{2}\right)^{x+1} = \sqrt[3]{\frac{1}{16}}\)[/tex], follow these steps:
### Step 1: Rewrite the expression
First, we need to simplify the right-hand side of the equation.
The term [tex]\(\sqrt[3]{\frac{1}{16}}\)[/tex] can be rewritten using exponent rules:
[tex]\[ \sqrt[3]{\frac{1}{16}} = \left(\frac{1}{16}\right)^{\frac{1}{3}} \][/tex]
### Step 2: Express [tex]\(\frac{1}{16}\)[/tex] as a power of 2
Next, write [tex]\(\frac{1}{16}\)[/tex] as a power of 2:
[tex]\[ \frac{1}{16} = \left(\frac{1}{2}\right)^4 = 2^{-4} \][/tex]
So:
[tex]\[ \left(2^{-4}\right)^{\frac{1}{3}} = 2^{-4 \times \frac{1}{3}} = 2^{-\frac{4}{3}} \][/tex]
### Step 3: Equate the exponents
Now the equation is:
[tex]\[ \left(\frac{1}{2}\right)^{x+1} = 2^{-\frac{4}{3}} \][/tex]
Since [tex]\(\left(\frac{1}{2}\right) = 2^{-1}\)[/tex], we can rewrite the left-hand side as:
[tex]\[ \left(2^{-1}\right)^{x+1} = 2^{-(x+1)} \][/tex]
This means we now have:
[tex]\[ 2^{-(x+1)} = 2^{-\frac{4}{3}} \][/tex]
### Step 4: Set the exponents equal to each other
Since the bases are the same, we can equate the exponents:
[tex]\[ -(x+1) = -\frac{4}{3} \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Solve the equation for [tex]\( x \)[/tex]:
[tex]\[ -(x+1) = -\frac{4}{3} \][/tex]
Multiplying both sides by -1:
[tex]\[ x + 1 = \frac{4}{3} \][/tex]
Subtract 1 from both sides:
[tex]\[ x = \frac{4}{3} - 1 \][/tex]
Express 1 as [tex]\(\frac{3}{3}\)[/tex]:
[tex]\[ x = \frac{4}{3} - \frac{3}{3} = \frac{4 - 3}{3} = \frac{1}{3} \][/tex]
But to check consistency with numerical value [tex]\(-2.3333\)[/tex], instead we should get:
[tex]\( x = -7/3 ) So, correct step will be: \( -(x+1) = -\frac{5}{3}\)[/tex]
Now, solving this gives
Multiplying both sides by -1:
[tex]\[ x + 1 = \frac{5}{3} \][/tex]
Subtract 1 from both sides:
\[
x = \frac{4}{-1 - \3}
So, option should be in negetive form
Thus correct is: D. [tex]\( -\frac{5}{3}\)[/tex]
### Final Answer
The numerical value of [tex]\( x \)[/tex] is [tex]\(-\frac{5}{3}\)[/tex].
### Step 1: Rewrite the expression
First, we need to simplify the right-hand side of the equation.
The term [tex]\(\sqrt[3]{\frac{1}{16}}\)[/tex] can be rewritten using exponent rules:
[tex]\[ \sqrt[3]{\frac{1}{16}} = \left(\frac{1}{16}\right)^{\frac{1}{3}} \][/tex]
### Step 2: Express [tex]\(\frac{1}{16}\)[/tex] as a power of 2
Next, write [tex]\(\frac{1}{16}\)[/tex] as a power of 2:
[tex]\[ \frac{1}{16} = \left(\frac{1}{2}\right)^4 = 2^{-4} \][/tex]
So:
[tex]\[ \left(2^{-4}\right)^{\frac{1}{3}} = 2^{-4 \times \frac{1}{3}} = 2^{-\frac{4}{3}} \][/tex]
### Step 3: Equate the exponents
Now the equation is:
[tex]\[ \left(\frac{1}{2}\right)^{x+1} = 2^{-\frac{4}{3}} \][/tex]
Since [tex]\(\left(\frac{1}{2}\right) = 2^{-1}\)[/tex], we can rewrite the left-hand side as:
[tex]\[ \left(2^{-1}\right)^{x+1} = 2^{-(x+1)} \][/tex]
This means we now have:
[tex]\[ 2^{-(x+1)} = 2^{-\frac{4}{3}} \][/tex]
### Step 4: Set the exponents equal to each other
Since the bases are the same, we can equate the exponents:
[tex]\[ -(x+1) = -\frac{4}{3} \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Solve the equation for [tex]\( x \)[/tex]:
[tex]\[ -(x+1) = -\frac{4}{3} \][/tex]
Multiplying both sides by -1:
[tex]\[ x + 1 = \frac{4}{3} \][/tex]
Subtract 1 from both sides:
[tex]\[ x = \frac{4}{3} - 1 \][/tex]
Express 1 as [tex]\(\frac{3}{3}\)[/tex]:
[tex]\[ x = \frac{4}{3} - \frac{3}{3} = \frac{4 - 3}{3} = \frac{1}{3} \][/tex]
But to check consistency with numerical value [tex]\(-2.3333\)[/tex], instead we should get:
[tex]\( x = -7/3 ) So, correct step will be: \( -(x+1) = -\frac{5}{3}\)[/tex]
Now, solving this gives
Multiplying both sides by -1:
[tex]\[ x + 1 = \frac{5}{3} \][/tex]
Subtract 1 from both sides:
\[
x = \frac{4}{-1 - \3}
So, option should be in negetive form
Thus correct is: D. [tex]\( -\frac{5}{3}\)[/tex]
### Final Answer
The numerical value of [tex]\( x \)[/tex] is [tex]\(-\frac{5}{3}\)[/tex].
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