Simplify the following expressions:
1) \(\Large\frac{\left(3a^2b\right)^{\normalsize{4}}}{a^{-3}}\cdot\left(\frac{1}{a^6b^{-1}}\right)^{-4}\cdot\normalsize\left(4a^{-2}b^{-5}\right)^{-3}\)
2) \(\Large\frac{\left(4x^3z^7\right)^3\left(-x^2y^4\right)^{11}}{\left(-3x^8\right)^2\left(x^{-9}y^5\right)^{-3}}\)
3) \(\Large\left(\frac{\left(5x^4y^{-5}\right)^{-3}}{\left(10x^6y^{-4}\right)^{-3}}\right)^{-2}\)
4) \(\large\left(x\strut{^{\frac{-4}{6}}}y\strut{^{\frac{1}{4}}}\right)^{-5}\left(x^{-7}y\strut{^{\frac{3}{4}}}\right)^{\frac{4}{-7}}\)
5) \(\Large\frac{\left(x^4y^{-1}z\right)^6}{4x^9z^{-4}}\div\frac{y^3z^{-6}}{\left(-2x^2y^{-5}\right)^4}\)
6) \(\left(4x^4y^{-2}\right)^2\cdot\left(\Large\frac{3x^3y^{-2}}{12x^{-8}y^0}\right)^{-6}\)
7) Let \(\Large\frac{28^{12}\cdot14^6}{8^3}\large=2^x\cdot y^y\) where \(x\) and \(y\) are integers. Find \(x\) and \(y\).
8) Arrange from least to greatest: \(5^{102},\ 3^{204},\ 2^{255}\).
9) Let \(w=2^a\). Write \((2\cdot2^a)^3-2^{(6+a)}\) in terms of \(w\).
10) Determine the value of \( \frac{k}{2w} \) when \( \left( \frac{1}{4} \right)^{2x-3} - \left( \frac{1}{16} \right)^{x-2} + 2^{3-4x}= k (2)^{wx} \) for all real values of \(x \).
Solution Bank
1) \(\Large\frac{\left(3a^2b\right)^{\normalsize{4}}}{a^{-3}}\cdot\left(\frac{1}{a^6b^{-1}}\right)^{-4}\cdot\normalsize\left(4a^{-2}b^{-5}\right)^{-3}\)
2) \(\Large\frac{\left(4x^3z^7\right)^3\left(-x^2y^4\right)^{11}}{\left(-3x^8\right)^2\left(x^{-9}y^5\right)^{-3}}\)
3) \(\Large\left(\frac{\left(5x^4y^{-5}\right)^{-3}}{\left(10x^6y^{-4}\right)^{-3}}\right)^{-2}\)
4) \(\large\left(x\strut{^{\frac{-4}{6}}}y\strut{^{\frac{1}{4}}}\right)^{-5}\left(x^{-7}y\strut{^{\frac{3}{4}}}\right)^{\frac{4}{-7}}\)
5) \(\Large\frac{\left(x^4y^{-1}z\right)^6}{4x^9z^{-4}}\div\frac{y^3z^{-6}}{\left(-2x^2y^{-5}\right)^4}\)
6) \(\left(4x^4y^{-2}\right)^2\cdot\left(\Large\frac{3x^3y^{-2}}{12x^{-8}y^0}\right)^{-6}\)
7) Let \(\Large\frac{28^{12}\cdot14^6}{8^3}\large=2^x\cdot y^y\) where \(x\) and \(y\) are integers. Find \(x\) and \(y\).
8) Arrange from least to greatest: \(5^{102},\ 3^{204},\ 2^{255}\).
9) Let \(w=2^a\). Write \((2\cdot2^a)^3-2^{(6+a)}\) in terms of \(w\).
10) Determine the value of \( \frac{k}{2w} \) when \( \left( \frac{1}{4} \right)^{2x-3} - \left( \frac{1}{16} \right)^{x-2} + 2^{3-4x}= k (2)^{wx} \) for all real values of \(x \).
Solution Bank