EXPONENTS AND POWERS simplify,
a)
[tex]( \frac{3}{8} ) { }^{ - 2} \times ( \frac{4}{5} ) {}^{ - 3} [/tex]
b)
[tex]( \frac{ - 2}{7} ) {}^{ - 4} \times ( \frac{ - 7}{5} ) { }^{2} [/tex]
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EXPONENTS AND POWERS simplify,
a)
[tex]( \frac{3}{8} ) { }^{ - 2} \times ( \frac{4}{5} ) {}^{ - 3} [/tex]
b)
[tex]( \frac{ - 2}{7} ) {}^{ - 4} \times ( \frac{ - 7}{5} ) { }^{2} [/tex]
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Answer:
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Let's simplify these expressions:
a)
\[
\left(3^8\right)^{-2} \times \left(4^5\right)^{-3} \div \left(8^3\right)^{-2} \times \left(5^4\right)^{-3}
\]
We can use the property of exponents which states that \((a^b)^c = a^{b \cdot c}\) to simplify the expression:
\[
3^{8 \cdot (-2)} \times 4^{5 \cdot (-3)} \div 8^{3 \cdot (-2)} \times 5^{4 \cdot (-3)}
\]
Now, calculate the exponents:
\[
3^{-16} \times 4^{-15} \div 8^{-6} \times 5^{-12}
\]
To simplify further, we can use the rule \(a^{-n} = \frac{1}{a^n}\):
\[
\frac{1}{3^{16}} \times \frac{1}{4^{15}} \div \frac{1}{8^6} \times \frac{1}{5^{12}}
\]
Now, use the rule \(a^{-n} = \frac{1}{a^n}\) to convert them back to positive exponents:
\[
\frac{1}{3^{16}} \times \frac{1}{4^{15}} \div \frac{1}{8^6} \times \frac{1}{5^{12}}
\]
b)
\[
\left(\left(-\frac{2}{7}\right)^{-4}\right) \times \left(\left(-\frac{7}{5}\right)^2\right) \div \left(\left(7^{-2}\right)\right) \times \left(\left(5^{-7}\right)^2\right)
\]
Again, use the property of exponents \((a^b)^c = a^{b \cdot c}\) to simplify:
\[
\left(\left(-\frac{2}{7}\right)^{-4 \cdot 1}\right) \times \left(\left(-\frac{7}{5}\right)^{2 \cdot 1}\right) \div \left(\left(7^{-2 \cdot 1}\right)\right) \times \left(\left(5^{-7 \cdot 2}\right)\right)
\]
Now calculate the exponents:
\[
\left(-\frac{2}{7}\right)^{-4} \times \left(-\frac{7}{5}\right)^2 \div 7^{-2} \times 5^{-14}
\]
Again, use the rule \(a^{-n} = \frac{1}{a^n}\):
\[
\frac{1}{\left(-\frac{2}{7}\right)^4} \times \frac{1}{\left(-\frac{7}{5}\right)^2} \div \frac{1}{7^2} \times \frac{1}{5^{14}}
\]
Now, convert them back to positive exponents:
\[
\frac{1}{\left(-\frac{2}{7}\right)^4} \times \frac{1}{\left(-\frac{7}{5}\right)^2} \div 7^2 \times 5^{14}
\]
These are the simplified forms of the given expressions.