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Section 1: Determina o valor de cada uma das potências

1.1 $5^2$

$5^2 = 5 \times 5 = 25$

1.2 $(-2)^4$

$(-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16$

1.3 $(-1)^3$

$(-1)^3 = (-1) \times (-1) \times (-1) = -1$

1.4 $(-3)^3$

$(-3)^3 = (-3) \times (-3) \times (-3) = -27$

1.5 $6^0$

$6^0 = 1 \quad \text{(any number raised to 0 is 1, except 0)}$

1.6 $3^{-2}$

$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

1.7 $(-2)^{-4}$

$(-2)^{-4} = \frac{1}{(-2)^4} = \frac{1}{16}$

1.8 $\left(\frac{-1}{2}\right)^{-3}$

$\left(\frac{-1}{2}\right)^{-3} = \frac{1}{\left(\frac{-1}{2}\right)^3} = \frac{1}{\frac{-1}{8}} = -8$

1.9 $\left(\frac{-2}{3}\right)^{-2}$

$\left(\frac{-2}{3}\right)^{-2} = \frac{1}{\left(\frac{-2}{3}\right)^2} = \frac{1}{\frac{4}{9}} = \frac{9}{4}$

1.10 $(2^0 / 3)^{-1}$

$2^0 = 1, \quad \text{so } \frac{2^0}{3} = \frac{1}{3}.$ $\left(\frac{1}{3}\right)^{-1} = 3$

Section 2: Calcula o valor de cada uma das potências

2.1 $8^3$

$8^3 = 8 \times 8 \times 8 = 512$

2.2 $4^{\frac{3}{2}}$

$4^{\frac{3}{2}} = \left(\sqrt{4}\right)^3 = 2^3 = 8$

Section 3: Escreve na forma de potência de base 2

3.1 $(2^4)^3$

$(2^4)^3 = 2^{4 \times 3} = 2^{12}$

3.2 $\left(\frac{1}{8}\right)^{-21}$

$\frac{1}{8} = 2^{-3}, \quad \text{so } \left(\frac{1}{8}\right)^{-21} = (2^{-3})^{-21} = 2^{-3 \times -21} = 2^{63}$

3.3 $\sqrt[3]{2^4}$

$\sqrt[3]{2^4} = 2^{\frac{4}{3}}$

3.4 $\sqrt[5]{\left(\frac{2}{3}\right)^3}$

$\sqrt[5]{\left(\frac{2}{3}\right)^3} = \left(\frac{2}{3}\right)^{\frac{3}{5}}$

Section 4: Calcula o valor numérico das expressões

4.1 $3^{-26} \times 3^{28}$

$3^{-26} \times 3^{28} = 3^{-26 + 28} = 3^2 = 9$

4.2 $(-2)^{36} \times (-2)^{-37}$

$(-2)^{36} \times (-2)^{-37} = (-2)^{36 - 37} = (-2)^{-1} = \frac{-1}{2}$

4.3 $5^{-26} + 5^{-28}$

$5^{-26} = \frac{1}{5^{26}}, \quad 5^{-28} = \frac{1}{5^{28}}.$ $\text{Adding these yields } \frac{1}{5^{26}} + \frac{1}{5^{28}} = \frac{25}{5^{28}}.$

4.4 $(-3)^{-2} + (-3)^{-2}$

$(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}.$ $\frac{1}{9} + \frac{1}{9} = \frac{2}{9}.$

4.5 $36^x \times (3^x)^6$

$36 = 6^2 = (3^2)^2 = 3^4.$ $36^x \times (3^x)^6 = (3^4)^x \times (3^6)^x = 3^{4x} \times 3^{6x} = 3^{10x}.$

4.6 $\left(\frac{1}{3}\right)^x \times \left(\frac{1}{3}\right)^6 + 6^{-4}$

$\left(\frac{1}{3}\right)^x \times \left(\frac{1}{3}\right)^6 = \left(\frac{1}{3}\right)^{x+6} = 3^{-(x+6)}.$ $6^{-4} = \frac{1}{6^4}.$

Would you like further explanations for any part?

Here are 5 related questions for practice:

1. Simplify $(5^3)^4$.
2. Rewrite $\sqrt[4]{16}$ as a power of 2.
3. Compute $\left(\frac{1}{4}\right)^{-2}$.
4. Simplify $(-1)^n$ where $n$ is odd.
5. Find $2^{-5} \times 2^{3}$.

Tip: When dealing with negative exponents, remember $a^{-n} = \frac{1}{a^n}$.