Answer :
To solve the expression \(\frac{1}{2^{-2} \cdot x^{-3} \cdot y^5}\) for \(x = 2\) and \(y = -4\), let's evaluate it step by step.
### Step 1: Evaluate \(2^{-2}\)
First, compute the value of \(2^{-2}\):
[tex]\[2^{-2} = \frac{1}{2^2} = \frac{1}{4}\][/tex]
So, \(2^{-2} = 0.25\).
### Step 2: Evaluate \(x^{-3}\)
Next, compute the value of \(x^{-3}\) when \(x = 2\):
[tex]\[2^{-3} = \frac{1}{2^3} = \frac{1}{8}\][/tex]
So, for \(x = 2\), \(x^{-3} = 0.125\).
### Step 3: Evaluate \(y^5\)
Now, compute the value of \(y^5\) when \(y = -4\):
[tex]\[(-4)^5 = (-4) \cdot (-4) \cdot (-4) \cdot (-4) \cdot (-4) = -1024\][/tex]
So, for \(y = -4\), \(y^5 = -1024\).
### Step 4: Combine the parts
Now, substitute these values into the original expression:
[tex]\[ \frac{1}{2^{-2} \cdot x^{-3} \cdot y^5} = \frac{1}{0.25 \cdot 0.125 \cdot (-1024)} \][/tex]
### Evaluate the denominator expression
First, multiply the values inside the denominator:
[tex]\[ 0.25 \cdot 0.125 = 0.03125 \][/tex]
Then,
[tex]\[ 0.03125 \cdot (-1024) = -32 \][/tex]
So the expression becomes:
[tex]\[ \frac{1}{-32} = -0.03125 \][/tex]
Thus, the value of the expression \(\frac{1}{2^{-2} \cdot x^{-3} \cdot y^5}\) when \(x=2\) and \(y=-4\) is \(-0.03125\).
Therefore, the correct answer is:
[tex]\(\boxed{-\frac{1}{32}}\)[/tex], which is equivalent to [tex]\(-0.03125\)[/tex].
### Step 1: Evaluate \(2^{-2}\)
First, compute the value of \(2^{-2}\):
[tex]\[2^{-2} = \frac{1}{2^2} = \frac{1}{4}\][/tex]
So, \(2^{-2} = 0.25\).
### Step 2: Evaluate \(x^{-3}\)
Next, compute the value of \(x^{-3}\) when \(x = 2\):
[tex]\[2^{-3} = \frac{1}{2^3} = \frac{1}{8}\][/tex]
So, for \(x = 2\), \(x^{-3} = 0.125\).
### Step 3: Evaluate \(y^5\)
Now, compute the value of \(y^5\) when \(y = -4\):
[tex]\[(-4)^5 = (-4) \cdot (-4) \cdot (-4) \cdot (-4) \cdot (-4) = -1024\][/tex]
So, for \(y = -4\), \(y^5 = -1024\).
### Step 4: Combine the parts
Now, substitute these values into the original expression:
[tex]\[ \frac{1}{2^{-2} \cdot x^{-3} \cdot y^5} = \frac{1}{0.25 \cdot 0.125 \cdot (-1024)} \][/tex]
### Evaluate the denominator expression
First, multiply the values inside the denominator:
[tex]\[ 0.25 \cdot 0.125 = 0.03125 \][/tex]
Then,
[tex]\[ 0.03125 \cdot (-1024) = -32 \][/tex]
So the expression becomes:
[tex]\[ \frac{1}{-32} = -0.03125 \][/tex]
Thus, the value of the expression \(\frac{1}{2^{-2} \cdot x^{-3} \cdot y^5}\) when \(x=2\) and \(y=-4\) is \(-0.03125\).
Therefore, the correct answer is:
[tex]\(\boxed{-\frac{1}{32}}\)[/tex], which is equivalent to [tex]\(-0.03125\)[/tex].
