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Sagot :
Certainly! Let's address each expression one by one in detail.
#### Expression f)
[tex]\[ \frac{2^7}{2^5} \][/tex]
When dividing expressions with the same base, you subtract the exponents:
[tex]\[ 2^{7-5} = 2^2 = 4 \][/tex]
So, the result is:
[tex]\[ \frac{2^7}{2^5} = 4 \][/tex]
#### Expression 8)
[tex]\[ 5^2 - 4^2 \][/tex]
This is a difference of squares. The formula for the difference of squares is:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Here:
[tex]\[ 5^2 - 4^2 = (5 - 4)(5 + 4) = 1 \cdot 9 = 9 \][/tex]
So, the result is:
[tex]\[ 5^2 - 4^2 = 9 \][/tex]
#### Expression 2a)
[tex]\[ \frac{2^{-1} - (-2)^2 + (-2) \cdot 1}{2^2 + 2^{-2}} \][/tex]
First, calculate the numerator:
[tex]\[ 2^{-1} = \frac{1}{2} \][/tex]
[tex]\[ (-2)^2 = 4 \][/tex]
[tex]\[ (-2) \cdot 1 = -2 \][/tex]
So, the numerator is:
[tex]\[ \frac{1}{2} - 4 - 2 = \frac{1}{2} - 6 = -\frac{11}{2} \][/tex]
Next, calculate the denominator:
[tex]\[ 2^2 = 4 \][/tex]
[tex]\[ 2^{-2} = \frac{1}{4} \][/tex]
So, the denominator is:
[tex]\[ 4 + \frac{1}{4} = \frac{16}{4} + \frac{1}{4} = \frac{17}{4} \][/tex]
Now compute the whole fraction:
[tex]\[ \frac{-\frac{11}{2}}{\frac{17}{4}} = \frac{-11}{2} \cdot \frac{4}{17} = -\frac{44}{34} = -\frac{22}{17} \approx -1.2941176470588236 \][/tex]
So, the result is approximately:
[tex]\[ \frac{2^{-1} - (-2)^2 + (-2) \cdot 1}{2^2 + 2^{-2}} \approx -1.2941176470588236 \][/tex]
#### Expression 2c)
[tex]\[ \left(\frac{-1}{2}\right)^2 \cdot \left(\frac{1}{2}\right)^3 \][/tex]
First, compute each term:
[tex]\[ \left(\frac{-1}{2}\right)^2 = \frac{1}{4} \][/tex]
[tex]\[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \][/tex]
Now multiply:
[tex]\[ \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32} = 0.03125 \][/tex]
So, the result is:
[tex]\[ \left(\frac{-1}{2}\right)^2 \cdot \left(\frac{1}{2}\right)^3 = 0.03125 \][/tex]
#### Expression 2b)
[tex]\[ \frac{3^2 - 3^{-2}}{3^2 + 3^{-2}} \][/tex]
First, compute the numerator:
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ 3^{-2} = \frac{1}{9} \][/tex]
So, the numerator is:
[tex]\[ 9 - \frac{1}{9} = \frac{81}{9} - \frac{1}{9} = \frac{80}{9} \][/tex]
Next, compute the denominator:
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ 3^{-2} = \frac{1}{9} \][/tex]
So, the denominator is:
[tex]\[ 9 + \frac{1}{9} = \frac{81}{9} + \frac{1}{9} = \frac{82}{9} \][/tex]
Now compute the whole fraction:
[tex]\[ \frac{\frac{80}{9}}{\frac{82}{9}} = \frac{80}{82} = \frac{40}{41} \approx 0.9756097560975611 \][/tex]
So, the result is approximately:
[tex]\[ \frac{3^2 - 3^{-2}}{3^2 + 3^{-2}} \approx 0.9756097560975611 \][/tex]
#### Expression 2d)
[tex]\[ \left[\left(-\frac{1}{2}\right)^2\right]^3 \][/tex]
First, compute the inner term:
[tex]\[ \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \][/tex]
Now raise it to the power of 3:
[tex]\[ \left(\frac{1}{4}\right)^3 = \frac{1}{64} = 0.015625 \][/tex]
So, the result is:
[tex]\[ \left[\left(-\frac{1}{2}\right)^2\right]^3 = 0.015625 \][/tex]
In summary, the detailed solutions are:
1. [tex]\( \frac{2^7}{2^5} = 4 \)[/tex]
2. [tex]\( 5^2 - 4^2 = 9 \)[/tex]
3. [tex]\( \frac{2^{-1} - (-2)^2 + (-2) \cdot 1}{2^2 + 2^{-2}} \approx -1.2941176470588236 \)[/tex]
4. [tex]\( \left(\frac{-1}{2}\right)^2 \cdot \left(\frac{1}{2}\right)^3 = 0.03125 \)[/tex]
5. [tex]\( \frac{3^2 - 3^{-2}}{3^2 + 3^{-2}} \approx 0.9756097560975611 \)[/tex]
6. [tex]\( \left[\left(-\frac{1}{2}\right)^2\right]^3 = 0.015625 \)[/tex]
#### Expression f)
[tex]\[ \frac{2^7}{2^5} \][/tex]
When dividing expressions with the same base, you subtract the exponents:
[tex]\[ 2^{7-5} = 2^2 = 4 \][/tex]
So, the result is:
[tex]\[ \frac{2^7}{2^5} = 4 \][/tex]
#### Expression 8)
[tex]\[ 5^2 - 4^2 \][/tex]
This is a difference of squares. The formula for the difference of squares is:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Here:
[tex]\[ 5^2 - 4^2 = (5 - 4)(5 + 4) = 1 \cdot 9 = 9 \][/tex]
So, the result is:
[tex]\[ 5^2 - 4^2 = 9 \][/tex]
#### Expression 2a)
[tex]\[ \frac{2^{-1} - (-2)^2 + (-2) \cdot 1}{2^2 + 2^{-2}} \][/tex]
First, calculate the numerator:
[tex]\[ 2^{-1} = \frac{1}{2} \][/tex]
[tex]\[ (-2)^2 = 4 \][/tex]
[tex]\[ (-2) \cdot 1 = -2 \][/tex]
So, the numerator is:
[tex]\[ \frac{1}{2} - 4 - 2 = \frac{1}{2} - 6 = -\frac{11}{2} \][/tex]
Next, calculate the denominator:
[tex]\[ 2^2 = 4 \][/tex]
[tex]\[ 2^{-2} = \frac{1}{4} \][/tex]
So, the denominator is:
[tex]\[ 4 + \frac{1}{4} = \frac{16}{4} + \frac{1}{4} = \frac{17}{4} \][/tex]
Now compute the whole fraction:
[tex]\[ \frac{-\frac{11}{2}}{\frac{17}{4}} = \frac{-11}{2} \cdot \frac{4}{17} = -\frac{44}{34} = -\frac{22}{17} \approx -1.2941176470588236 \][/tex]
So, the result is approximately:
[tex]\[ \frac{2^{-1} - (-2)^2 + (-2) \cdot 1}{2^2 + 2^{-2}} \approx -1.2941176470588236 \][/tex]
#### Expression 2c)
[tex]\[ \left(\frac{-1}{2}\right)^2 \cdot \left(\frac{1}{2}\right)^3 \][/tex]
First, compute each term:
[tex]\[ \left(\frac{-1}{2}\right)^2 = \frac{1}{4} \][/tex]
[tex]\[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \][/tex]
Now multiply:
[tex]\[ \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32} = 0.03125 \][/tex]
So, the result is:
[tex]\[ \left(\frac{-1}{2}\right)^2 \cdot \left(\frac{1}{2}\right)^3 = 0.03125 \][/tex]
#### Expression 2b)
[tex]\[ \frac{3^2 - 3^{-2}}{3^2 + 3^{-2}} \][/tex]
First, compute the numerator:
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ 3^{-2} = \frac{1}{9} \][/tex]
So, the numerator is:
[tex]\[ 9 - \frac{1}{9} = \frac{81}{9} - \frac{1}{9} = \frac{80}{9} \][/tex]
Next, compute the denominator:
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ 3^{-2} = \frac{1}{9} \][/tex]
So, the denominator is:
[tex]\[ 9 + \frac{1}{9} = \frac{81}{9} + \frac{1}{9} = \frac{82}{9} \][/tex]
Now compute the whole fraction:
[tex]\[ \frac{\frac{80}{9}}{\frac{82}{9}} = \frac{80}{82} = \frac{40}{41} \approx 0.9756097560975611 \][/tex]
So, the result is approximately:
[tex]\[ \frac{3^2 - 3^{-2}}{3^2 + 3^{-2}} \approx 0.9756097560975611 \][/tex]
#### Expression 2d)
[tex]\[ \left[\left(-\frac{1}{2}\right)^2\right]^3 \][/tex]
First, compute the inner term:
[tex]\[ \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \][/tex]
Now raise it to the power of 3:
[tex]\[ \left(\frac{1}{4}\right)^3 = \frac{1}{64} = 0.015625 \][/tex]
So, the result is:
[tex]\[ \left[\left(-\frac{1}{2}\right)^2\right]^3 = 0.015625 \][/tex]
In summary, the detailed solutions are:
1. [tex]\( \frac{2^7}{2^5} = 4 \)[/tex]
2. [tex]\( 5^2 - 4^2 = 9 \)[/tex]
3. [tex]\( \frac{2^{-1} - (-2)^2 + (-2) \cdot 1}{2^2 + 2^{-2}} \approx -1.2941176470588236 \)[/tex]
4. [tex]\( \left(\frac{-1}{2}\right)^2 \cdot \left(\frac{1}{2}\right)^3 = 0.03125 \)[/tex]
5. [tex]\( \frac{3^2 - 3^{-2}}{3^2 + 3^{-2}} \approx 0.9756097560975611 \)[/tex]
6. [tex]\( \left[\left(-\frac{1}{2}\right)^2\right]^3 = 0.015625 \)[/tex]
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