Powers and roots

Simplify

$$\frac{5^7\cdot 5^{-4}}{5^2}.$$

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Simplify

$$\frac{{\left(2^3\right)}^5}{4^6}.$$

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Simplify, for $a,b\ne 0$,

$${\left(\frac{a^{2}b}{a{b}^3}\right)}^3.$$

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Simplify

$$\sqrt{48}-\sqrt{27}+\sqrt{12}.$$

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Determine for which real $x$ the following expression is defined:

$$\sqrt{x-4}+\frac{1}{\sqrt{10-x}}.$$

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Determine which of $2^{60}$ and $5^{20}$ is larger.

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Rationalize

$$\frac{5}{\sqrt{10}}.$$

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Rationalize

$$\frac{2}{\sqrt{6}+2}.$$

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Rationalize

$$\frac{3}{\sqrt[3]{9}}.$$

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Order from smallest to largest

$$\sqrt[3]{2},\sqrt[4]{3},\sqrt{2}.$$

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