求 e ^{{In x – 2ln y}}的简单表达式

数学不仅与数字有关，而且与涉及数字和变量的不同计算有关。这就是基本上被称为代数的东西。代数被定义为涉及由数字、运算符和变量组成的数学表达式的计算的表示。数字可以是 0 到 9，运算符是数学运算符，如 +、-、×、÷、指数等，变量如 x、y、z 等。

指数和幂

指数和幂是数学计算中使用的基本运算符，指数用于简化涉及多次自乘的复杂计算，自乘基本上是数字与自身相乘。例如，7 × 7 × 7 × 7 × 7，可以简单地写成 7 ⁵ 。这里，7 是基值，5 是指数，值为 16807。11 × 11 × 11，可以写为 11 ³ ，这里，11 是基值，3 是 11 的指数或幂。 11 ³是 1331。

指数被定义为一个数字的幂，它乘以自身的次数。如果表达式写成 cx ^y其中 c 是常数，c 将是系数，x 是底数，y 是指数。如果一个数 p 乘以 n 次，n 将是 p 的指数。它将被写为

p × p × p × p … n 次 = p ⁿ

指数的基本规则

为了求解指数表达式以及其他数学运算，为指数定义了一些基本规则，例如，如果有两个指数的乘积，则可以简化以使计算更容易，称为乘积规则，让我们看一下指数的一些基本规则，

• 乘积规则 ⇢ a ⁿ × a ^m = a ^{n + m}
• 商规则 ⇢ a ⁿ / a ^m = a ^{n – m}
• 幂律 ⇢ (a ⁿ ) ^m = a ^{n × m}或^m √a ⁿ = a ^n/m
• 负指数规则 ⇢ a ^-m = 1/a ^m
• 零规则 ⇢ a ⁰ = 1
• 一条规则 ⇢ a ¹ = a

对数的基本规则

在数学中，对数被定义为指数的倒数。对数计算重复乘法中相同因子的出现次数。例如，10000 = 10 × 10 × 10 × 10 = 10 ⁴ ，在这种情况下，10000 的以 10 为底的对数是 4。以下是一些最基本的对数规则，

• 乘积规则 ⇢ log _b mn = log _b m + log _b n
• 商规则 ⇢ log _b m/n = log _b m – log _b n
• 幂律 ⇢ log _b m ^p = plog _b m
• 等式规则 ⇢ 如果 log _b m = log _b n。那么，m = n

求 e ^{{In x – 2ln y}}的简单表达式

解决方案：

As it is clearly seen, the entire problem statement is asking for a simplification using exponent rules and basic logarithmic rules, looking at the expression e^{{In x-2ln y}}, it is observed that the power rule of logarithms can be easily applied to this expression,

Power Rule ⇢ log_b m^p = plog_b m

e^{{In x-2ln y}} = e^{{In x – ln y2}}

Apply the quotient rule of exponents.

Quotient Rule ⇢ aⁿ / a^m = a^{n – m}

e^{{In x-ln y2}} = e^lnx / e^lny2

= x / y²

Therefore, x / y² is the value obtained.

类似问题

问题1：什么是10m ⁵ (3m ⁷ )？

解决方案：

As it is clearly seen, the entire problem statement is asking for a simplification using exponent rules, looking at the expression 10m⁵(3m⁷), it is observed that the product rule of exponents can be easily applied to this expression,

Step 1: Remove the parenthesis and write terms with their exponents.

10m⁵(3m⁷) = 10m⁵ × 3m⁷

Step 2: Apply the product rule of exponents.

Product Rule ⇢ aⁿ × a^m = a^{n + m}

10m⁵ × 3m⁷ = 10 × 3m^{(5 + 7)}

Therefore, 30m¹² is the value obtained.

问题 2：化简 (x ¹⁰ )(x ² )

解决方案：

As it is clearly seen, the entire problem statement is asking for a simplification using exponent rules, looking at the expression (x¹⁰)(x²), it is observed that the product rule of exponents can be easily applied to this expression,

Product Rule ⇢ aⁿ × a^m = a^{n + m}

x¹⁰ × x² = x^{(10 + 2)}

= x¹²

Therefore, x¹² is the value obtained.

问题 3：化简 e ^{{6In x – 5ln y}}

解决方案：

As it is clearly seen, the entire problem statement is asking for a simplification using exponent rules and basic logarithmic rules, looking at the expression e^{{6In x-5ln y}}, it is observed that the power rule of logarithms can be easily applied to this expression,

Power Rule ⇢ log_b m^p = plog_b m

e^{{6In x – 5ln y}} = e^{{In x6 – ln y5}}

Apply the quotient rule of exponents.

Quotient Rule ⇢ aⁿ / a^m = a^{n – m}

e^{{In x6 – ln y5}} = elnx⁶ / elny²

= x⁶ / y⁵

Therefore, x⁶ / y⁵ is the value obtained.