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Sagot :
Sure, let us solve the given problems step-by-step:
### Part f:
[tex]\[ \left(-\frac{7}{8}\right)^4 \cdot \left[\left(-\frac{7}{8}\right)^3\right]^2 \div \left(-\frac{7}{8}\right)^8 \][/tex]
First, we use exponent rules to simplify this expression. Remember that [tex]\(\left(a^m\right)^n = a^{mn}\)[/tex] and [tex]\(a^m \div a^n = a^{m-n}\)[/tex].
1. Simplify [tex]\(\left[\left(-\frac{7}{8}\right)^3\right]^2\)[/tex]:
[tex]\[ \left[\left(-\frac{7}{8}\right)^3\right]^2 = \left(-\frac{7}{8}\right)^{3 \cdot 2} = \left(-\frac{7}{8}\right)^6 \][/tex]
2. Now our expression is:
[tex]\[ \left(-\frac{7}{8}\right)^4 \cdot \left(-\frac{7}{8}\right)^6 \div \left(-\frac{7}{8}\right)^8 \][/tex]
3. Combine the exponents in the numerator:
[tex]\[ \left(-\frac{7}{8}\right)^4 \cdot \left(-\frac{7}{8}\right)^6 = \left(-\frac{7}{8}\right)^{4+6} = \left(-\frac{7}{8}\right)^{10} \][/tex]
4. Now, we have:
[tex]\[ \left(-\frac{7}{8}\right)^{10} \div \left(-\frac{7}{8}\right)^8 \][/tex]
5. Simplify by subtracting the exponents:
[tex]\[ \left(-\frac{7}{8}\right)^{10-8} = \left(-\frac{7}{8}\right)^2 \][/tex]
So, the result for part f is:
[tex]\[ \left(-\frac{7}{8}\right)^2 = \frac{49}{64} \][/tex]
### Part g:
[tex]\[ \left[\left(\frac{9}{4}\right)^2 \cdot \left(\frac{9}{4}\right)^6\right]^3 \div \left(\frac{4}{9}\right)^{-20} \][/tex]
Again, use exponent rules to simplify.
1. Combine the exponents inside the brackets:
[tex]\[ \left(\left(\frac{9}{4}\right)^2 \cdot \left(\frac{9}{4}\right)^6\right) = \left(\frac{9}{4}\right)^{2+6} = \left(\frac{9}{4}\right)^8 \][/tex]
2. Now the expression is:
[tex]\[ \left[\left(\frac{9}{4}\right)^8\right]^3 \div \left(\frac{4}{9}\right)^{-20} \][/tex]
3. Simplify the expression in the numerator:
[tex]\[ \left(\left(\frac{9}{4}\right)^8\right)^3 = \left(\frac{9}{4}\right)^{8 \cdot 3} = \left(\frac{9}{4}\right)^{24} \][/tex]
4. Simplify the denominator:
[tex]\[ \left(\frac{4}{9}\right)^{-20} = \left(\frac{9}{4}\right)^{20} \][/tex]
5. Now, we have:
[tex]\[ \left(\frac{9}{4}\right)^{24} \div \left(\frac{9}{4}\right)^{20} = \left(\frac{9}{4}\right)^{24-20} = \left(\frac{9}{4}\right)^4 \][/tex]
So, the result for part g is:
[tex]\[ \left(\frac{9}{4}\right)^4 = \frac{9^4}{4^4} = \frac{6561}{256} \][/tex]
In conclusion, we have:
Part f: [tex]\(\frac{49}{64}\)[/tex]
Part g: [tex]\(\frac{6561}{256}\)[/tex]
### Part f:
[tex]\[ \left(-\frac{7}{8}\right)^4 \cdot \left[\left(-\frac{7}{8}\right)^3\right]^2 \div \left(-\frac{7}{8}\right)^8 \][/tex]
First, we use exponent rules to simplify this expression. Remember that [tex]\(\left(a^m\right)^n = a^{mn}\)[/tex] and [tex]\(a^m \div a^n = a^{m-n}\)[/tex].
1. Simplify [tex]\(\left[\left(-\frac{7}{8}\right)^3\right]^2\)[/tex]:
[tex]\[ \left[\left(-\frac{7}{8}\right)^3\right]^2 = \left(-\frac{7}{8}\right)^{3 \cdot 2} = \left(-\frac{7}{8}\right)^6 \][/tex]
2. Now our expression is:
[tex]\[ \left(-\frac{7}{8}\right)^4 \cdot \left(-\frac{7}{8}\right)^6 \div \left(-\frac{7}{8}\right)^8 \][/tex]
3. Combine the exponents in the numerator:
[tex]\[ \left(-\frac{7}{8}\right)^4 \cdot \left(-\frac{7}{8}\right)^6 = \left(-\frac{7}{8}\right)^{4+6} = \left(-\frac{7}{8}\right)^{10} \][/tex]
4. Now, we have:
[tex]\[ \left(-\frac{7}{8}\right)^{10} \div \left(-\frac{7}{8}\right)^8 \][/tex]
5. Simplify by subtracting the exponents:
[tex]\[ \left(-\frac{7}{8}\right)^{10-8} = \left(-\frac{7}{8}\right)^2 \][/tex]
So, the result for part f is:
[tex]\[ \left(-\frac{7}{8}\right)^2 = \frac{49}{64} \][/tex]
### Part g:
[tex]\[ \left[\left(\frac{9}{4}\right)^2 \cdot \left(\frac{9}{4}\right)^6\right]^3 \div \left(\frac{4}{9}\right)^{-20} \][/tex]
Again, use exponent rules to simplify.
1. Combine the exponents inside the brackets:
[tex]\[ \left(\left(\frac{9}{4}\right)^2 \cdot \left(\frac{9}{4}\right)^6\right) = \left(\frac{9}{4}\right)^{2+6} = \left(\frac{9}{4}\right)^8 \][/tex]
2. Now the expression is:
[tex]\[ \left[\left(\frac{9}{4}\right)^8\right]^3 \div \left(\frac{4}{9}\right)^{-20} \][/tex]
3. Simplify the expression in the numerator:
[tex]\[ \left(\left(\frac{9}{4}\right)^8\right)^3 = \left(\frac{9}{4}\right)^{8 \cdot 3} = \left(\frac{9}{4}\right)^{24} \][/tex]
4. Simplify the denominator:
[tex]\[ \left(\frac{4}{9}\right)^{-20} = \left(\frac{9}{4}\right)^{20} \][/tex]
5. Now, we have:
[tex]\[ \left(\frac{9}{4}\right)^{24} \div \left(\frac{9}{4}\right)^{20} = \left(\frac{9}{4}\right)^{24-20} = \left(\frac{9}{4}\right)^4 \][/tex]
So, the result for part g is:
[tex]\[ \left(\frac{9}{4}\right)^4 = \frac{9^4}{4^4} = \frac{6561}{256} \][/tex]
In conclusion, we have:
Part f: [tex]\(\frac{49}{64}\)[/tex]
Part g: [tex]\(\frac{6561}{256}\)[/tex]
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