求解表达式： (7/3) ^(-3)

在测量中，当我们必须计算正方形的面积时，我们只是简单地并排相乘。这导致一个数字与相同的数字相乘。假设有 n 行，并且在每一行中，总共有“n”个元素。对于元素总数的计算，我们将总行数乘以一行中的元素数。这就是我们如何引入指数的概念。

例如，

当我们必须一次又一次地添加相同的数字时，我们引入了乘法这个术语。

比如，5 + 5 + 5 + 5 = 4 × 5 = 20

同样，如果我们必须一次又一次地乘以相同的数字，我们就会引入术语指数。

例如，2 × 2 × 2 × 2 × 2 = 2 ⁵

因此，在本文中，我们将讨论指数和幂以及它的不同规则如何用于解决问题。

Exponent is defined as the number of times a number is multiplied by itself.

Suppose, 5 is multiplied 4 times.

5 × 5 × 5 × 5 = 5⁴

Here, the number which is multiplied is called as base i.e. 5, and the number of times it multiplied is called as power or index or exponent, i.e. 4.

We will read it as ‘five raised to the power four’ or ‘five to the power four’ or ‘fourth power of five’.

5⁴ is called as exponential form or exponential notation of 625.

指数规则

规则1：当数字乘以不同的幂但相同的基数时，它们的幂相加。

a^m × aⁿ = a^(m+n)

Example, 2³ × 2² = 2⁽³⁺²⁾ = 2⁵

规则 2：当数字乘以不同的基数但相同的幂时，它们的基数相乘。

aⁿ × bⁿ = (ab)ⁿ

Example : 2³ × 3³ = (2 × 3 )³ = 6³

规则 3：当一个数字具有其幂的幂时，幂就会成倍增加。

(a^m)ⁿ = a^(m×n)

Example: (2³)⁴ = 2^(3×4) = 2¹²

规则 4：当数以不同的幂除以相同的基数时，则从分子的幂中减去其分母的幂，或者从被除数的幂中减去除数的幂。

a^m ÷ aⁿ = a^(m-n)

Example: 2⁶ ÷ 2⁵ = 2^(6-5) 

规则 5：当数字以不同的底数除以相同的幂时，则它们的底数被除而幂保持不变。

a^m ÷ b^m = (a ÷ b)^m

Example: 3² ÷ 4² = (3×4)²

规则 6：一个数的负幂代表它自己的倒数。

a^(-n) = 1/aⁿ

Example: 5^(-2) = 1/5²

规则 7：任何数的零次方等于 1。

a⁰ = 1

Example: 23⁰ = 1

概念：

1. 要解决指数问题，请尝试找出应用上述规则中的哪一个。
2. 使用属性并以最简单的形式减少问题。
3. 进行适当的计算并得到最终答案。

求解表达式： (7/3) ^(-3)

解决方案：

We have to solve (7/3)^(-3).

Base of the above problem is (7/3) and power is {-3}.

Negative power is introduced in rule number 6, i.e., a^(-n) = 1/aⁿ

Negative power of any number change the base into its reciprocal.

Reciprocal of 7/3 = 3/7

Since their power will remain same, 

(7/3)^(-3) = (3/7)³

Now expand (3/7)³ using reverse of rule number 5.

= (3³)/(7³)

= (3×3×3)/(7×7×7)

= 27/343

So, the final answer of (7/3)^(-3) is 27/343.

类似问题

问题 1：求解 (2/3) ^(-2) 。这遵循什么规则？

解决方案：

We have to solve (2/3)^(-2).

Base of the above problem is (2/3) and power is {-2}.

Negative power is introduced in rule number 6, i.e., a^(-n) = 1/aⁿ

Negative power of any number change the base into its reciprocal.

Reciprocal of 2/3 = 3/2

Since their power will remain same,

(2/3)^(-2) = (3/2)²

Now expand (3/2)² using reverse of rule number 5.

= (3²)/(2²)

= (3×3)/(2×2)

= 9/4

So, the final answer of  (2/3)^(-2) is 9/4.

问题 2：求解 (1/3) ^(-3) 。这遵循什么规则？

解决方案：

We have to solve (1/3)^(-3).

Base of the above problem is (1/3) and power is {-3}.

Negative power is introduced in rule number 6, i.e., a^(-n) = 1/aⁿ

Negative power of any number change the base into its reciprocal.

So, the reciprocal of 1/3 is 3.

Since their power will remain same,

(1/3)^(-3) = (3)³

Now expand (3)³.

= 3 × 3 × 3

= 27

So, the final answer of (1/3)^(-3) is 27.