Surds and Indices

If $9^{{\mathrm{x}}^2}=9^{2\mathrm{x}-1}$, find the value of ${\left(\mathrm{x}+1\right)}^2.$

Find the value of expression:

$\frac{{\left({\displaystyle \frac{2}{5}}\right)}^0-{\left({\displaystyle \frac{1}{10}}\right)}^{-1}}{{\left({\displaystyle \frac{3}{2^3}}\right)}^{-1}\left({\displaystyle \frac{27}{8}}\right)+{\left(-{\displaystyle \frac{1}{3}}\right)}^{-1}}$

Simplify the following:

$\frac{4^{\mathrm{n}}(10\times 2{)}^{\mathrm{m}-1}\times (10+2{)}^{\mathrm{m}-\mathrm{n}}\times {\left(10+5\right)}^{\mathrm{m}+\mathrm{n}-2}}{{\left(4^2\right)}^{\mathrm{m}}\times 5^{2\mathrm{m}+\mathrm{n}}\times {\left(3^2\right)}^{\mathrm{m}-1}}$

Simplify the following:

$\left(2^4+2^3+2^{-3}+2^{-2}\right)$

If $4^{\mathrm{x}+2}+4^{\mathrm{x}-2}=1028$

Then, $\mathrm{x}= ?$

Simplify:

 ${\left(\mathrm{a}+\frac{1}{\mathrm{b}}\right)}^{\mathrm{m}}{\left(\mathrm{a}-\frac{1}{\mathrm{b}}\right)}^{\mathrm{n}}÷{\left(\mathrm{b}-\frac{1}{\mathrm{a}}\right)}^{\mathrm{n}}{\left(\mathrm{b}+\frac{1}{\mathrm{a}}\right)}^{\mathrm{m}}$

What will come in place of the question mark (?) in the given question?

${81}^{2.5} \times 9^{4.5}÷ 3^{4.8} = 9^{?}$

What should come in place of the question mark (?) in the following questions?
$(6{)}^4÷(36{)}^3\times 216=6^{(?-2)}$
 

What will come in place of the question-mark (?) in the following question ?

$$\sqrt { 9216 } \times \sqrt { 2601 } -{ \left( 43 \right) }^{ 2 }=\sqrt { ? } +{ \left( 55 \right) }^{ 2 }$$

What will come in place of the question-mark (?) in the following question ?

$\left[\left(7\sqrt{2}+9\sqrt{2}\right)\times \left(4\sqrt{2}+11\sqrt{2}\right)\right]-257=?$

What will come in place of the question-mark (?) in the following question ?

$$\frac { 14 }{ 39 } \times \frac { 117 }{ 154 } \div \frac { 78 }{ 88 } +\frac { 15 }{ 78 } =?$$

What will come in place of the question-mark $(?)$ in the following question ?

${\left(17.3\right)}^2+{\left(37.2\right)}^2-{\left(32.6\right)}^2=?+437.21$

What will come in place of the question-mark (?) in the following question ?

If a-b= 3 and a² + b² = 29, then ab=?