指数法则

初步定义

自然指数

$$\Large \boxed{a^n = \begin{cases} a & \text{若 } n = 1 \\ \underbrace{a \cdot a \cdot \ldots \cdot a}_{n \text{ 次}} & \text{若 } n \in \mathbb{N}, n \geq 2 \end{cases}}$$

负底数幂的符号规则：

$$(-b)^{2n} = +b^{2n}; \, \forall n \in \mathbb{Z}^+$$

$$(-b)^{2n+1} = -b^{2n+1}; \, \forall n \in \mathbb{Z}^+$$

示例

• $3^4 = \underbrace{3 \cdot 3 \cdot 3 \cdot 3}_{4 \text{ 次}} = 81$
• $\left(\dfrac{1}{2}\right)^3 = \underbrace{\dfrac{1}{2} \cdot \dfrac{1}{2} \cdot \dfrac{1}{2}}_{3 \text{ 次}} = \dfrac{1}{8}$
• $(-2)^5 = \underbrace{(-2) \cdot (-2) \cdot (-2) \cdot (-2) \cdot (-2)}_{5 \text{ 次}} = -32$
• $x^3 = \underbrace{x \cdot x \cdot x}_{3 \text{ 次}}$
• $7^1 = 7 \quad \text{(根据定义，若 } n = 1\text{)}$
• $(-1)^6 = \underbrace{(-1) \cdot (-1) \cdot (-1) \cdot (-1) \cdot (-1) \cdot (-1)}_{6 \text{ 次}} = 1$

零指数

$$\Large \boxed{a^0 = 1}$$

$$\forall a \in \mathbb{R} \setminus \{0\}$$

示例

• $5^0 = 1$
• $(-7)^0 = 1$
• $(3.14)^0 = 1$
• $\left(\frac{2}{3}\right)^0 = 1$
• $(x^2 + 1)^0 = 1 \quad \text{对于任意 } x \in \mathbb{R}$
• $(a \cdot b \cdot c)^0 = 1 \quad \text{若 } a, b, c \neq 0$

负指数

$$\Large \boxed{a^{-n} = \frac{1}{a^n}}$$

$$\forall a \in \mathbb{R} \setminus \{0\}, n \in \mathbb{N}$$

示例

• $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$
• $5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}$
• $(-4)^{-3} = \dfrac{1}{(-4)^3} = \dfrac{1}{-64} = -\dfrac{1}{64}$
• $\left(\dfrac{1}{3}\right)^{-2} = \dfrac{1}{\left(\dfrac{1}{3}\right)^2} = \dfrac{1}{\dfrac{1}{9}} = 9$
• $x^{-4} = \dfrac{1}{x^4} \quad \text{若 } x \neq 0$
• $(2a)^{-1} = \dfrac{1}{2a} \quad \text{若 } a \neq 0$

分数指数

$$\Large \boxed{a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m}$$

$$\forall n \in \mathbb{N}, n \geq 2$$

示例

• $8^{2/3} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$
• $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$
• $27^{1/3} = \sqrt[3]{27} = 3$
• $4^{3/2} = (\sqrt{4})^3 = 2^3 = 8 \quad \text{(此处 } \sqrt[2]{\ } = \sqrt{\ }\text{)}$
• $x^{5/2} = \sqrt{x^5} = (\sqrt{x})^5 \quad \text{若 } x \geq 0$
• $\left(\dfrac{1}{9}\right)^{1/2} = \sqrt{\dfrac{1}{9}} = \dfrac{1}{3}$

幂运算

性质

同底数幂相乘

$$\Large \boxed{a^m \cdot a^n = a^{m+n}}$$

$$a \in \mathbb{R}, m, n \in \mathbb{N}$$

示例

• $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$
• $x^5 \cdot x^2 = x^{5+2} = x^7$
• $(-3)^2 \cdot (-3)^6 = (-3)^{2+6} = (-3)^8 = 6561$
• $a^4 \cdot a^1 = a^{4+1} = a^5$
• $\left(\dfrac{1}{2}\right)^3 \cdot \left(\dfrac{1}{2}\right)^5 = \left(\dfrac{1}{2}\right)^{3+5} = \left(\dfrac{1}{2}\right)^8 = \dfrac{1}{256}$
• $y^{n} \cdot y^{7} = y^{n+7}$

幂的幂

$$\Large \boxed{(a^m)^n = a^{m \cdot n} \quad}$$

$$a \in \mathbb{R}, m, n \in \mathbb{N}$$

$$\boxed{\left(((a^m)^n)^r\right)^s = a^{m \cdot n \cdot r \cdot s}}$$

示例

• $(2^3)^4 = 2^{3 \cdot 4} = 2^{12} = 4096$
• $(x^2)^5 = x^{2 \cdot 5} = x^{10}$
• $((-3)^4)^2 = (-3)^{4 \cdot 2} = (-3)^8 = 6561$
• $(a^3)^1 = a^{3 \cdot 1} = a^3$
• $\left(\left(\dfrac{1}{2}\right)^2\right)^3= \left(\dfrac{1}{2}\right)^{2 \cdot 3} = \left(\dfrac{1}{2}\right)^6 = \dfrac{1}{64}$
• $(y^m)^7 = y^{m \cdot 7} = y^{7m}$

积的幂

$$\Large \boxed{(a \cdot b)^n = a^n \cdot b^n}$$

$$a, b \in \mathbb{R}, n \in \mathbb{N}$$

$$\boxed{(x^a \cdot y^b)^n = x^{a \cdot n} \cdot y^{b \cdot n}}$$

示例

• $(3 \cdot 4)^2 = 3^2 \cdot 4^2 = 9 \cdot 16 = 144$
• $(2x)^3 = 2^3 \cdot x^3 = 8x^3$
• $(-5 \cdot 2)^4 = (-5)^4 \cdot 2^4 = 625 \cdot 16 = 10000$
• $(ab)^5 = a^5 \cdot b^5$
• $\left(\dfrac{1}{2} \cdot y\right)^3 = \left(\dfrac{1}{2}\right)^3 \cdot y^3 = \dfrac{1}{8}y^3$
• $(xy^2)^4 = x^4 \cdot (y^2)^4 = x^4 \cdot y^8$

同底数幂相除

$$\Large \boxed{\frac{a^m}{a^n} = a^{m-n}}$$

$$m, n \in \mathbb{N}, m \geq n, a \in \mathbb{R} \setminus \{0\}$$

示例

• $\dfrac{5^7}{5^3} = 5^{7-3} = 5^4 = 625$
• $\dfrac{x^8}{x^5} = x^{8-5} = x^3$
• $\dfrac{(-2)^6}{(-2)^2} = (-2)^{6-2} = (-2)^4 = 16$
• $\dfrac{a^{10}}{a^{10}} = a^{10-10} = a^0 = 1 \quad \text{(若 } a \neq 0\text{)}$
• $\dfrac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$
• $\dfrac{y^{n+4}}{y^n} = y^{(n+4)-n} = y^4$

商的幂

$$\Large \boxed{\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}}$$

$$n \in \mathbb{N}, b \in \mathbb{R} \setminus \{0\}$$

示例

• $\left(\dfrac{2}{3}\right)^4 = \dfrac{2^4}{3^4} = \dfrac{16}{81}$
• $\left(\dfrac{5}{2}\right)^3 = \dfrac{5^3}{2^3} = \dfrac{125}{8}$
• $\left(\dfrac{x}{4}\right)^2 = \dfrac{x^2}{4^2} = \dfrac{x^2}{16} \quad \text{若 } x \in \mathbb{R}$
• $\left(\dfrac{-3}{5}\right)^3 = \dfrac{(-3)^3}{5^3} = \dfrac{-27}{125}$
• $\left(\dfrac{a}{b}\right)^5 = \dfrac{a^5}{b^5} \quad \text{若 } b \neq 0$
• $\left(\dfrac{1}{x}\right)^n = \dfrac{1^n}{x^n} = \dfrac{1}{x^n} \quad \text{若 } x \neq 0$

分数的负指数

$$\Large \boxed{\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n = \frac{b^n}{a^n}}$$

$$a, b \neq 0$$

示例

• $\left(\dfrac{2}{3}\right)^{-4} = \left(\dfrac{3}{2}\right)^4 = \dfrac{3^4}{2^4} = \dfrac{81}{16}$
• $\left(\dfrac{5}{x}\right)^{-2} = \left(\dfrac{x}{5}\right)^2 = \dfrac{x^2}{25} \quad \text{若 } x \neq 0$
• $\left(\dfrac{1}{4}\right)^{-3} = \left(\dfrac{4}{1}\right)^3 = 4^3 = 64$
• $\left(\dfrac{a}{b}\right)^{-1} = \dfrac{b}{a} \quad \text{若 } a, b \neq 0$
• $\left(\dfrac{-2}{7}\right)^{-3} = \left(\dfrac{7}{-2}\right)^3 = \dfrac{7^3}{(-2)^3} = \dfrac{343}{-8} = -\dfrac{343}{8}$
• $\left(\dfrac{x}{y}\right)^{-5} = \dfrac{y^5}{x^5} \quad \text{若 } x, y \neq 0$

连续指数

$$\Large \boxed{x^{a^{b^c}} = x^{a^m}= x^n= z}$$

示例

• $2^{2^{3^1}} = 2^{2^3} = 2^8 = 256$
• $3^{2^{2^2}} = 3^{2^4} = 3^{16} = 43,\!046,\!721$
• $5^{1^{4^2}} = 5^{1^{16}} = 5^1 = 5$
• $10^{3^{1^5}} = 10^{3^1} = 10^3 = 1,\!000$
• $2^{3^{2^1}} = 2^{3^2} = 2^9 = 512$
• $4^{2^{3^0}} = 4^{2^1} = 4^2 = 16$

绝对值

$$\Large \boxed{ \sqrt[2k]{x^{2k}} = |x| = \begin{cases} x & \text{若： } x > 0 \\ 0 & \text{若： } x = 0 \\ -x & \text{若： } x < 0 \end{cases}}$$

实数中的根式

符号规则：

$$\sqrt[2n+1]{(+)} = (+)$$

$$\sqrt[2n+1]{(-)} = (-)$$

$$\sqrt[2n]{(+)} = (+)$$

$$\sqrt[2n]{(-)} = \text{在 }\mathbb{R}\text{ 中无定义}$$

性质

积的根

$$\Large \boxed{\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}}$$

若 $n$ 为偶数，则 $a \geq 0$ 且 $b \geq 0$。

示例

• $\sqrt{4 \cdot 9} = \sqrt{4} \cdot \sqrt{9} = 2 \cdot 3 = 6$
• $\sqrt[3]{8 \cdot 27} = \sqrt[3]{8} \cdot \sqrt[3]{27} = 2 \cdot 3 = 6$
• $\sqrt[4]{16 \cdot 81} = \sqrt[4]{16} \cdot \sqrt[4]{81} = 2 \cdot 3 = 6$
• $\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y} \quad \text{若 } x \geq 0,\ y \geq 0$
• $\sqrt[5]{a^5 \cdot b^{10}} = \sqrt[5]{a^5} \cdot \sqrt[5]{b^{10}} = a \cdot b^2$
• $\sqrt[3]{-8 \cdot 64} = \sqrt[3]{-8} \cdot \sqrt[3]{64} = (-2) \cdot 4 = -8$

商的根

$$\Large \boxed{\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}}$$

$$b \neq 0$$

若 $n$ 为偶数，则 $a \geq 0$ 且 $b > 0$。

示例

• $\sqrt{\dfrac{9}{16}} = \dfrac{\sqrt{9}}{\sqrt{16}} = \dfrac{3}{4}$
• $\sqrt[3]{\dfrac{27}{125}} = \dfrac{\sqrt[3]{27}}{\sqrt[3]{125}} = \dfrac{3}{5}$
• $\sqrt[4]{\dfrac{1}{81}} = \dfrac{\sqrt[4]{1}}{\sqrt[4]{81}} = \dfrac{1}{3}$
• $\sqrt[5]{\dfrac{x^5}{32}} = \dfrac{\sqrt[5]{x^5}}{\sqrt[5]{32}} = \dfrac{x}{2} \quad \text{若 } x \in \mathbb{R}$
• $\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}} \quad \text{若 } a \geq 0,\ b > 0$
• $\sqrt[3]{\dfrac{-8}{27}} = \dfrac{\sqrt[3]{-8}}{\sqrt[3]{27}} = \dfrac{-2}{3}$

根的根

$$\Large \boxed{\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \cdot n]{a}}$$

$$m, n \in \mathbb{N}$$

若 $m \cdot n$ 为偶数，则 $a \geq 0$。

$$\boxed{\sqrt[m]{\sqrt[n]{\sqrt[s]{\sqrt[r]{a}}}} = \sqrt[m \cdot n \cdot s \cdot r]{a}}$$

示例

• $\sqrt[2]{\sqrt[3]{8}} = \sqrt[6]{8} = \sqrt[6]{2^3} = 2^{3/6} = 2^{1/2} = \sqrt{2}$
• $\sqrt[3]{\sqrt[4]{x}} = \sqrt[12]{x} \quad \text{对于 } x \geq 0$
• $\sqrt{\sqrt{16}} = \sqrt[2]{\sqrt[2]{16}} = \sqrt[4]{16} = 2$
• $\sqrt[5]{\sqrt{a}} = \sqrt[10]{a} \quad \text{若 } a \geq 0$
• $\sqrt[4]{\sqrt[3]{64}} = \sqrt[12]{64} = \sqrt[12]{2^6} = 2^{6/12} = 2^{1/2} = \sqrt{2}$
• $\sqrt{\sqrt{\sqrt{x}}} = \sqrt[2]{\sqrt[2]{\sqrt[2]{x}}} = \sqrt[8]{x} \quad \text{对于 } x \geq 0$

根的幂

$$\Large \boxed{\left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}}$$

$$\boxed{\sqrt[ab]{x^{ac}} = \sqrt[b]{x^c} \quad \text{若 } ab \text{ 为偶数，则 } x \in \mathbb{R}_0^+}$$

示例

• $\left(\sqrt[3]{2}\right)^6 = \sqrt[3]{2^6} = \sqrt[3]{64} = 4$
• $\left(\sqrt{5}\right)^4 = \sqrt{5^4} = \sqrt{625} = 25$
• $\left(\sqrt[4]{3}\right)^2 = \sqrt[4]{3^2} = \sqrt[4]{9}$
• $\left(\sqrt[n]{x}\right)^3 = \sqrt[n]{x^3} \quad \text{若 } x \geq 0 \text{ 且 } n \text{ 为偶数}$
• $\left(\sqrt[5]{-32}\right)^3 = \sqrt[5]{(-32)^3} = \sqrt[5]{-32768} = -8$
• $\left(\sqrt{a^2 + b^2}\right)^2 = \sqrt{(a^2 + b^2)^2} = |a^2 + b^2| = a^2 + b^2$

连续根式

$$\sqrt[n]{x^{a}\cdot \sqrt[m]{x^{b}\cdot \sqrt[p]{x^c}}} = \sqrt[nmp]{x^{a \cdot m \cdot p + b \cdot p + c}}$$

$$\sqrt[n]{x^{a}\div \sqrt[m]{x^{b}\div \sqrt[p]{x^c}}} = \sqrt[nmp]{x^{a \cdot m \cdot p - b \cdot p + c}}$$

$$\sqrt[m]{x \cdot \sqrt[n]{y \cdot \sqrt[p]{z}}} = \sqrt[m]{x} \cdot \sqrt[m \cdot n]{y} \cdot \sqrt[m \cdot n \cdot p]{z}$$

$$\sqrt[m]{x^a \cdot \sqrt[n]{y^b \cdot \sqrt[p]{z^c}}} = \sqrt[m]{x^a} \cdot \sqrt[m \cdot n]{y^b} \cdot \sqrt[m \cdot n \cdot p]{z^c}$$

$$\sqrt[m]{x^{a}\div \sqrt[n]{y^{b}\div \sqrt[p]{z^{c}\div \sqrt[q]{w^{d}\div \sqrt[r]{v^{e}\div \sqrt[s]{u^f}}}}}} = \frac{x^{\frac{a}{m}} \cdot z^{\frac{c}{mnp}} \cdot v^{\frac{e}{mnpqr}} }{y^{\frac{b}{mn}} \cdot w^{\frac{d}{mnpq}} \cdot u^{\frac{f}{mnpqrs}}}$$

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辅助性质

$$x^m \cdot \sqrt[n]{y^p} = \sqrt[n]{x^{mn} \cdot y^p}$$

$$\sqrt[m]{x} \cdot \sqrt[n]{x} = \sqrt[mn]{x^{n+m}}$$

$$\frac{\sqrt[m]{x}}{\sqrt[n]{x}} = \sqrt[mn]{x^{n-m}}, \, x \neq 0$$

$$\sqrt[a]{\frac{\sqrt[b]{\cfrac{\sqrt[c]{\cfrac{1}{z}}}{y}}}{x}} = \sqrt[a]{\frac{1}{x}\cdot \sqrt[b]{\frac{1}{y}\cdot \sqrt[c]{\frac{1}{z}}}}$$

$$\frac{\sqrt[m]{a\cdot \sqrt[n]{b\cdot \sqrt[p]{c}} } }{\sqrt[m]{d\cdot \sqrt[n]{e\cdot \sqrt[p]{f}} } } = \sqrt[m]{\frac{a}{d}\cdot \sqrt[n]{\frac{b}{e}\cdot \sqrt[p]{\frac{c}{f}}}}$$

$$\sqrt[m]{x^{ \sqrt[n]{y^{ \sqrt[p]{z}}}}} = \sqrt[m]{x}^{\sqrt[n]{y}^{\sqrt[p]{z}}}$$

特殊情况

1. 含有特定数量根式的表达式

$$\underbrace{\sqrt[n]{ \sqrt[n]{ \sqrt[n]{ \dots \sqrt[n]{x}}}}}_{\text{m 个根式}} = \sqrt[n^m]{x}$$

$$\underbrace{\sqrt[n]{x \cdot \sqrt[n]{x \cdot \sqrt[n]{x \cdot ...\cdot \sqrt[n]{x}}}}}_{\text{m 个根式}} = \sqrt[n^m]{x^{\cfrac{n^m - 1}{n-1}}}$$

$$\underbrace{\sqrt[n]{x\div \sqrt[n]{x\div \sqrt[n]{x\div ... \cdot \div \sqrt[n]{x}}}}}_{''m'' \text{ 个根式}} = \begin{cases} \sqrt[n^m]{x^{\cfrac{n^m - 1}{n + 1}}} & \text{若 'm' 为偶数}, \\ \sqrt[n^m]{x^{\cfrac{n^m + 1}{n + 1}}} & \text{若 'm' 为奇数}. \end{cases}$$

2. 含有无穷根式的表达式

$$\sqrt{x\cdot(x+1)+ \sqrt{x\cdot(x+1)+ \sqrt{x\cdot(x+1)+ \dots } }} = x+1$$

$$\sqrt{x\cdot(x+1)- \sqrt{x\cdot(x+1)- \sqrt{x\cdot(x+1)- \dots } }} = x$$

$$\sqrt[n]{x\cdot \sqrt[n]{x\cdot \sqrt[n]{x\cdot...} }} = \sqrt[n-1]{x}$$

$$\sqrt[n]{x\div \sqrt[n]{x\div \sqrt[n]{x\div...} }} = \sqrt[n+1]{x}$$

$$\quad \sqrt[n]{a \cdot \sqrt[m]{a \cdot \sqrt[n]{a \cdot \sqrt[m]{a\cdot \dots} }}} = \sqrt[m \cdot n - 1]{a^{m+1}}$$

$$\quad \sqrt[n]{a \cdot \sqrt[m]{b \cdot \sqrt[n]{a \cdot \sqrt[m]{b\cdot \dots} }}} = \sqrt[m \cdot n - 1]{a^m \cdot b}$$

$$\quad \sqrt{a + \sqrt{a + \sqrt{a + \dots}}} = \frac{1 + \sqrt{4a + 1}}{2}$$

$$\quad \sqrt{a - \sqrt{a - \sqrt{a - \dots}}} = \frac{-1 + \sqrt{4a + 1}}{2}$$

3. 连续无穷指数

$$x^{x^{x^{{.}^{{.}^{{.}^{x^n}}}}}} = n \quad \rightarrow \quad x = \sqrt[n]{n}; \, x \neq 0$$

$$\sqrt[a]{b}^{\sqrt[a]{b}^{\sqrt[a]{b}^{\sqrt[a]{b}^{.^{.^{.}}}}}} = c \quad \rightarrow \quad b = c^{\,a}$$

Practice Problems

Level I

1.

Find the reduced value of $P$.

$$P = -3^2 + (-3)^2 - (-3)^0 + (-1)^0$$

A) 2
B) 0
C) -2
D) 1
E) -1

2.

Simplify the expression $R$.

$$R = \frac{6^5 \cdot 15^6}{3^{11} \cdot 10^5}$$

A) 3
B) 4
C) 5
D) 2
E) 1

3.

Simplify the expression $C$.

$$C = (2^{2^{2}})^{-1} - (-2)^{-2} \cdot 2^{-2}$$

A) 3
B) 4
C) 2
D) 8
E) 0

4.

Given that

$$A = \frac{5^{n+2} - 5^{n+1}}{5^{n+1}}$$

find the sum of the digits of $A^2$.

A) 5
B) 9
C) 11
D) 7
E) 8

5.

Simplify the expression $D$.

$$D = \frac{3^{n+1} + 3^{n+2} + 3^{n+3}}{3^{n-1} + 3^{n-2} + 3^{n-3}}$$

A) $3^4$
B) $3^5$
C) $3^7$
D) $3^6$
E) $3^8$

6.

Reduce the expression

$$\frac{2^{n+4}}{2^{n+3}} + \frac{5^{n+3}}{5^{n+1}} - \frac{3^{2-x}}{3^{1-x}}$$

A) 21
B) 4
C) 24
D) 10
E) 30

7.

Determine the reduced value of $M$.

$$M = \sqrt[3]{3^2} \cdot \sqrt[4]{2^5} \cdot \sqrt[3]{3} \cdot \sqrt[4]{2^{-1}}$$

A) 2
B) -4
C) -6
D) 6
E) 4

8.

Determine the reduced value of the following expression.

$$M = \frac{\sqrt{6} + \sqrt{12} + \sqrt{18} + \sqrt{24}}{\sqrt{3} + \sqrt{6} + \sqrt{9} + \sqrt{12}}$$

A) 2
B) $2^{\frac{1}{2}}$
C) $3^{\frac{1}{2}}$
D) $6^{\frac{1}{2}}$
E) $4^{\frac{1}{2}}$

9.

Determine the reduced value of $J$.

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

A) -2
B) 2
C) -1
D) 1
E) 3

10.

Given that $x^x = 7^{98}$, calculate the value of $\sqrt{\frac{x+1}{2}}$.

A) 4
B) 7
C) 8
D) 5
E) 6

Level II

1.

Given that $x^{x^5} = 2^{32}$, determine the value of $\sqrt[3]{2x^5}$.

A) 5
B) 32
C) 8
D) 2
E) 4

2.

Given the sequence $\{x_n\}$, such that
$x_1 = \sqrt{b}; x_2 = \sqrt{b\sqrt{b}}; x_3 = \sqrt{b\sqrt{b\sqrt{b}}}; \dots$
where $b$ is a positive real number, determine the value of $\frac{x_3 \cdot x_{10}}{(x_4 \cdot x_{11})^2}$.

A) $b^{-1/2}$
B) $b^{-2}$
C) $b^{-1/8}$
D) $b^{-3}$
E) $b^{-4}$

3.

Given that $x^{x^{\frac{1}{2}}} = \frac{1}{\sqrt{2}}$, determine the value of $x^{-1}$.

A) 64
B) 4
C) 16
D) 256
E) 512

4.

Given $a > 0$, calculate the value of $x$ in the following equality.

$$\sqrt[3]{a^{2x+1}} \cdot \sqrt[4]{a^{2-3x}} = \frac{1}{\sqrt{a^{x-1}}}$$

A) -3/5
B) -4/5
C) -1
D) -5/4
E) -5/2

5.

If $x^{\sqrt{x}}$ equals 2, determine the value of

$$\frac{(x+1)\sqrt{x}^{\sqrt{x}-1}}{\sqrt{x} + \frac{1}{\sqrt{x}}}$$

A) 1/2
B) 1
C) $\frac{\sqrt{2}}{2}$
D) $\sqrt{2}$
E) 2

6.

If $a$ and $b$ are coprime numbers, and $x^{\frac{a}{b}}$ is the result of reducing the expression

$$\frac{\sqrt{x \cdot \sqrt{x}} \cdot \sqrt{\sqrt{x\cdot x} }}{\sqrt{x \cdot \sqrt[3]{x^2}} }$$

then find the value of $b^2 - a^2$.
A) 144
B) 5
C) 169
D) 119
E) 36

7.

Given the numbers

$$A = 3 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}}$$

$$B = 2 - \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots}}}$$

Determine the value of $A \cdot B$.
A) -5
B) 12
C) 9
D) 1
E) -12

8.

Reduce the following expression

$$A = \frac{45^8 \times 75^{11} \times 225^7}{(3^{20} \times 5^{21})^2}$$

indicate the sum of the digits of $A$.

A) 10
B) 11
C) 12
D) 14
E) 9

9.

Simplify the following expression.

$$\frac{2^{-3n+6} + 2^{n+1} \cdot 4^{-2n+1}}{2 \cdot 8^{1-n}}$$

A) 3.0
B) 3.5
C) 4.5
D) 16.5
E) 7.5

10.

If $x^y = \frac{1}{2} \land y^x = 2$, calculate the value of $x^{y^{x+1}} + 1$.

A) 1/4
B) 1/2
C) 3/4
D) 5/4
E) 3/2

11.

If $5^x = m$ and $5^z = n$, find $(0.04)^{-x+2z}$

A) $m^2 \cdot n^{-4}$
B) $m^{1/2} \cdot n^{-4}$
C) $m^2 \cdot n^{-1/4}$
D) $m^{-2} \cdot n^4$
E) $m^2 \cdot n^4$

12.

Upon reducing the expression

$$\left(\frac{x^3y^3}{-x^4y^2}\right)^3 \cdot \left(\frac{x^3y}{-x^2y^2}\right)^5$$

$\left(\frac{y}{x}\right)^{-m}$ is obtained. Determine the value of $m^{m+1}$.

A) 2
B) 8
C) 4
D) -3
E) -2

13.

If $M = \left(\frac{a^{-2}-b^{-2}}{a^{-1}+b^{-1}}\right)^{-1}$ and $L = \left(\frac{a^{-1}-b^{-1}}{a^{-2}+b^{-2}}\right)^{-1}$; $(a \neq b)$, find the value of ML.

A) $\frac{1}{b^2-a^2}$
B) $\frac{ab}{a^2-b^2}$
C) $\frac{b+a}{a-b}$
D) $\frac{a^2-b^2}{ab}$
E) $\frac{b^2+a^2}{(b-a)^2}$

14.

If $2^{2^m}=8$, find the value of $\frac{4^{m+ 4^{m}}}{72}$.

A) $2^{10}$
B) $2^{16}$
C) $2^{12}$
D) $2^{18}$
E) $2^{15}$

15.

If $T = \sqrt[2m - 4n ]{\frac{5^{2m+1} 9^m \sqrt{5^{mn-1}}}{45^{2n} 5^m \sqrt{5^{mn+1}}}}$, find the positive difference of the values of x that satisfy the equation: $x^2 + \frac{T}{\sqrt{5}}x = 0$

A) 4
B) 2
C) 3
D) 5
E) 9

16.

Reduce:

$$T = \frac{\sqrt[mn]{(n^m \cdot m^{-n})^{-p}}}{(m^n)^{\frac{p-m}{mn}} (n^m)^{\frac{n-p}{mn}}}$$

A) $\frac{m}{n}$
B) $\frac{n}{m}$
C) 1
D) $\frac{1}{m}$
E) $\frac{1}{n}$

17.

If $\frac{a+b}{a-b} = \sqrt{\frac{a}{b} \sqrt{\frac{a}{b} \sqrt{\frac{a}{b} \cdots}}}$; $(a \neq b, a \neq -b)$, find the value of: $T = \frac{a-b}{b}$

A) 2
B) $\frac{\sqrt{2}}{2}$
C) $\sqrt{2}$
D) 1
E) 4

18.

If $a^2 + \frac{1}{a^2} = 7$; $(a > 0)$, find the value of $a^3 + \frac{1}{a^3}$.

A) 3
B) 18
C) 27
D) 21
E) 29

19.

If $(a\sqrt{2} - 2\sqrt{3})^2 = 30 - 12\sqrt{6}$, find the value of $a^2+a$.

A) 11
B) 12
C) 13
D) 14
E) 15

20.

Upon reducing the expression $x \cdot \sqrt[9]{x^2 \sqrt[5]{x^{3n} \cdot y^m}}$ the exponents of "x" and "y" are 2 and 4 respectively, find $m-3n$.

A) 180
B) 100
C) 145
D) 110
E) 180

Level III

1.

Determine the reduced equivalent of $P$.

$$P = 0,5 - \sqrt[4]{(-2)^2}^{-4^{2^{-5^0}}}$$

A) -1
B) -0,5
C) 0
D) 1
E) 0,25

2.

If $x + 2 = 23\sqrt{2x}$, calculate the value of

$$J = \frac{\sqrt{\sqrt{x} + \sqrt{2}}}{\sqrt[8]{2x}}$$

A) 4
B) $\sqrt{5}$
C) 5
D) 16
E) $\sqrt[8]{2}$

3.

Let $a; b \in \mathbb{R}^+$, such that $\frac{1}{a^2} + \frac{1}{b^2} = 1$
Determine the value of $S$.

$$S = \frac{\sqrt[a^2]{x^{b^2}} + \sqrt[b^2]{x^{a^2}}}{x^{a^2} + x^{b^2}}$$

A) $x^{-x}$
B) 1/x
C) x
D) $x^2$
E) 2

4.

Let $\{a; b; x\}$ be a set of elements different from unity, such that they satisfy $a^x = b^3 = x$.
Calculate the value of $x^{x^{-3^{-1}}}$ in terms of $a$ and $b$.

A) $ab^2$
B) $a^b$
C) $a^{b^2}$
D) $a^2\sqrt{b}$
E) $b^2\sqrt{a}$

5.

Indicate the final exponent of $x$ in the expression $J$.

$$J = \sqrt[3]{x \cdot \sqrt[5]{x^4 \cdot \sqrt[9]{x^{24} \cdot \sqrt[17]{x^{240} \dots}}}}$$

($m$ radicals)

A) $\frac{2^m+1}{2^m}$
B) $\frac{2^m-1}{2^m+1}$
C) $\frac{2^m+1}{2^m-1}$
D) $\frac{2^m}{2^m-1}$
E) $\frac{2^m}{2^m+1}$

6.

If the equalities hold

$$x^{x^9} = \sqrt{3^{\sqrt{3}}}; \quad y = x^{\left(\frac{1}{y^{y^x}}\right)}$$

calculate the value of $y^{3x}$.

A) 3
B) 2
C) $\sqrt{2}$
D) $\sqrt{3}$
E) 27

7.

Determine the value of $a$ given that

$$27^{81^{a+1}} = 3^{729^{a-2}}$$

A) 1/2
B) 5/3
C) 7/6
D) 17/2
E) -1/3

8.

Determine the reduced value of $M$.

$$M = \left( \frac{1}{5^2} \cdot \frac{1}{5^6} \cdot \frac{1}{5^{12}} \cdot \frac{1}{5^{20}} \cdot \dots \cdot \frac{1}{5^{420}} \right)^{\frac{21}{20}}$$

A) $5^{\frac{1}{2}}$
B) $5^{\frac{2013}{2012}}$
C) $5^{\frac{2012}{2013}}$
D) $5^0$
E) 5

9.

If upon reducing the expression

$$\left( \left( \frac{x^{-\frac{1}{n}} \cdot y^{\frac{m}{n}} }{x^{\frac{m+n}{n}} \cdot y^{\frac{1}{n}}} \right)^{-1} \div \left( \frac{x}{y} \right)^{\frac{m+n}{n}} \right)^n$$

the result is $x^n \cdot y^b$, calculate the value of $n^b + b^n$.

A) 1
B) 3
C) 5
D) 13
E) $2m$

10.

If $(x^{x^{x+3}})(x^{2x^{x+2}}) = a^{a^{\frac{1-a}{a}}}$
what is $\left(\frac{1}{a}\right)^{-\left(\frac{1}{a}\right)} \cdot x^{-x}$ equivalent to?

A) 1
B) $\sqrt{x}$
C) $x^{x+1}$
D) $x^2$
E) $x$

11.

After solving $x^{(x-1)^2} = 2x + 1; x > 0$, indicate the value of $x - \frac{1}{x}$.

A) 2
B) 4
C) 6
D) 8
E) 10

12.

Find the simplest value of N.

$$N=\left( 4^{\frac{\sqrt[5]{ 5 }}{\sqrt[3]{ 5 }}} \right)^{\left( \frac{\sqrt[5]{ 5 }+\sqrt[3]{ 5 }}{1+\sqrt[15]{ 25 }}\right)^{\left( \frac{3}{2}\right)^{-1}}}$$

A) 4
B) 25
C) 81
D) 9
E) 5

13.

If $a>b$, such that $\sqrt[a]{b}\sqrt[b]{a} = \sqrt[8]{2^{7\sqrt{2}}}$, calculate the value of $\frac{a+b}{a-b}$.

A) $5\sqrt{2}$
B) $2\sqrt{2}$
C) 3
D) 2
E) $\sqrt{2}$

14.

If

$$P = \frac{\left[(2+2^2)2^3-2^3\right]2^4-2^7}{\left\{(3^2-2)2^4-5 \cdot 2^4\right\}2^4 \cdot 3}$$

calculate $3P$.

A) 1
B) 2
C) 3
D) 1/3
E) -1

15.

After reducing the expression

$$\frac{(3^{4x-y})^y + (3^{4y-x})^x}{(3^y)^{-x^3} + (3^{-x})^{y^3}}$$

indicate the final exponent of 9 given that $x=\frac{1}{y}$.

A) 4
B) 2
C) 1
D) 3
E) 0

16.

Find $\left(\frac{1}{9}\right)^x$ if $32^{5^{3^x}} = \left[\left(\frac{1}{1024}\right)^{-2.5}\right]^5$.

A) 2
B) 9
C) 1/4
D) 4
E) 3

17.

In the following exponential equation

$$x^{x^3+2} = 5x^{5x}$$

calculate $x^4$.

A) 5
B) $\sqrt{5}$
C) 25
D) 4
E) 16

18.

Let:

$$\sqrt[xy]{z} \cdot \sqrt[xz]{y} \cdot \sqrt[yz]{x} = \sqrt{\sqrt{\sqrt{\sqrt[\frac{1}{2}]{2} \cdot (8\sqrt{2})^{\sqrt{2}}}}}$$

$$(\sqrt[n]{m} \cdot \sqrt[m]{n})^{m \cdot n} = \sqrt{ \underbrace{\sqrt{2^{\sqrt{2}} \cdot 2^{\sqrt{2}} \cdot 2^{\sqrt{2}} \dots 2^{\sqrt{2}}}}_{\text{"10 times"}} \cdot \sqrt[4]{ 256^{\sqrt{ 2 }} } }$$

One of the values of $xy+xz+yz+mn$ has the form $a+\sqrt{b}$, with $\{a;b\} \subset \mathbb{N}$. Find $a+b$.

A) 8,880
B) 88
C) 80
D) 8
E) 808

19.

Let:

$$A = ⟦ \frac{\left\{ \theta \left[ \theta^2 (\theta^3)^{-1} \right]^{-2} \right\}^{-3}}{\sqrt[2^{-3}]{ \sqrt[2^{-1}]{\sqrt[3^{-1}]{\frac{1}{\sqrt[3]{\sqrt{\theta}}}}} } } \cdot \theta^{-3^{0^{2^1}}} ⟧ ; \quad \text{with } \theta = \frac{\sqrt{5}+1}{4}$$

$$B=\sqrt{ \frac{\sqrt{ 7 }}{\sqrt{ \frac{1}{7} }}+\frac{\sqrt{ 7 \sqrt{ 7 }}}{\sqrt{ \sqrt{ \frac{1}{7} }}} +\frac{\sqrt{ 7 \sqrt{ 7 \sqrt{ 7 } } }}{\sqrt{ \sqrt{ \sqrt{ \frac{1}{7} } } }}+ \dots 343 \text{ terms} }$$

Where $⟦x⟧=n \leftrightarrow n \le x < n+1; n \in \mathbb{Z}$
Find the value of $A+B+AB$.

A) 50
B) 59
C) 97
D) 98
E) 99

20.

Indicate the value of $\sqrt[3]{x}^{1-7(3x+1)^{-1}}$ given that: $x^{x^{\sqrt[4]{x}+0.25}} = \left(\frac{1}{2}\right)^{\frac{\sqrt{2}}{2^{\sqrt{2}}}}$

A) 1
B) 2
C) 3
D) 4
E) $\sqrt[3]{3}$

21.

If $x = t^{\frac{1}{t-1}}$; $y = t^{\frac{t}{t-1}} / t>0 \land t \neq 1$ a relationship between $x$ and $y$ is.

A) $x^y = y^x$
B) $y^x = x^y$
C) $x^y = y^x$
D) $x^x = y^y$
E) A and B

22.

After solving: $2x+1 = \frac{3}{2}\left[\sqrt{\frac{1}{x}}\right]^{4x-1}$ if $x>0$ indicate the value of $\left(\frac{1}{x}\right)^{\frac{1}{x}}$

A) 16
B) 81
C) $\frac{1}{64}$
D) $\frac{1}{256}$
E) 256

23.

Given $x>0$ solve:

$$\sqrt[x-1]{1+\sqrt{2}} \cdot \sqrt[x+1]{\sqrt{2}-1} = \sqrt[16]{17+6\sqrt{8}}$$

A) 4
B) 3
C) -2
D) -3
E) 6

24.

Indicate the numerical value of: $M = \frac{1}{m}\left[\frac{x^m+\sqrt[m]{x}}{m+1}\right]^{m^2-1}$, $x = \sqrt[m+1]{m \sqrt[m-1]{m}}$
$m \in \mathbb{Z} \land m > 9$

A) m
B) 1
C) $m^m$
D) $\sqrt[m]{m}$
E) $\frac{1}{m}$

25.

Given: $A = \sqrt{2\sqrt{4\sqrt{8\sqrt{16\dots}}}}$; $B = \sqrt[3]{x\sqrt[5]{x^4\sqrt[9]{x^{24}\sqrt[17]{x^{240}\dots}}}}$
Calculate the approximate value of $\frac{A \cdot B}{x}$

A) 4
B) 2
C) 8
D) 1
E) 0

26.

Knowing that: $m = \sqrt[3]{x^2 \cdot \sqrt[3]{x^2 \dots}} \cdot \sqrt[4]{x^3 \cdot \sqrt[4]{x^3 \dots}} \cdot \sqrt[m+2]{x^{m+1} \cdot \sqrt[m+2]{x^{m+1} \dots}}$
Calculate the approximate value of "x" in terms of "m".

A) m
B) $m^m$
C) $m^{\frac{1}{m}}$
D) $\frac{1}{m}$
E) $m^2$

27.

Considering that: $A = k \sqrt[k]{\frac{\left(\frac{1}{n}\right)^{\frac{k}{2}}}{1 \cdot 2 \cdot 3 \dots k}} - \left(\frac{1}{\sqrt{n}} + \frac{1}{\sqrt{4n}} + \frac{1}{\sqrt{9n}} + \dots + \frac{1}{\sqrt{k^2n}}\right)$
Also "t" satisfies the condition $x^{x^t} = (tx)^x$, with $x=10^{1/9}$, calculate $t^2 \cdot \text{sgn}(A)$.

A) 10
B) 100
C) -10
D) -100
E) 0

28.

Calculate "n" if in the expression:

$$\left[ \dots \left[ \left[ (x^a)^{1/2} \cdot x^{-3/2} \right]^{1/2} \cdot x^{-3/2} \right]^{1/2} \dots x^{-3/2} \right]^{1/2}$$

('n' brackets)
the exponent of "x" after reducing the expression is 0.5; also $a = 2^{16}-3$.

A) 100
B) 110
C) 14
D) 2
E) 3

29.

Determine the value of $-x^{-x^{-x}+x^x} \cdot \left[ \frac{x^{-x^x} + x^{x^{-x}}}{x^{-x^{-x}} + x^{x^x}} \right]$ when $x = 2^{-100}$.

A) 0
B) 1
C) -1
D) $\frac{1}{64}$
E) $2^{-200}$

30.

Knowing that $\left(\frac{x}{\sqrt{2}}\right)^{\sqrt{2}+1} = \sqrt{2}$; also $y^{\frac{1}{z}} \cdot z^{\frac{1}{y}} = \sqrt[8]{2^{7\sqrt{2}}}$, calculate $zy(x^2)^{2^{-\frac{1}{2}}}$.

A) 8
B) 4
C) 2
D) 1
E) 16