Manipulating Exponential Expressions · 指数表达式的运算
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| quotient rule/ˈkwəʊʃənt ruːl/ | 除法法则 | chú fǎ fǎ zé |
| base/beɪs/ | 底数 | dǐ shù |
| product rule/ˈprɒdʌkt ruːl/ | 乘法法则 | chéng fǎ fǎ zé |
| power rule/ˈpaʊə ruːl/ | 幂法则 | mì fǎ zé |
| equivalent/ɪˈkwɪvələnt/ | 等价 | děng jià |
The rules for combining powers
- Once the variable lives in the exponent, a small set of rules lets you rearrange anything.
- Multiply powers, divide them, raise a power to a power — each has a shortcut.
- These shortcuts turn scary expressions into simple ones.
- They also let you rewrite an exponential in a more useful form.
合并幂的法则
- 一旦变量位于指数上,一小套法则就能让你随意重排任何式子。
- 幂相乘、相除、幂的幂——每一种都有捷径。
- 这些捷径把吓人的式子变得简单。
- 它们也能让你把指数函数改写成更有用的形式。
Product and quotient rules
- Multiplying powers of the same base 底数 adds the exponents: the product rule 乘法法则 $b^m \cdot b^n = b^{m+n}$.
- Dividing them subtracts the exponents: the quotient rule 除法法则 $\dfrac{b^m}{b^n} = b^{m-n}$.
- The base must be the same for either rule to apply.
- These match the intuition: $2^2 \cdot 2^3 = 4 \cdot 8 = 32 = 2^5$.
乘法法则与除法法则
- 同底数(base)的幂相乘,指数相加:乘法法则(product rule)$b^m \cdot b^n = b^{m+n}$。
- 相除则指数相减:除法法则(quotient rule)$\dfrac{b^m}{b^n} = b^{m-n}$。
- 两条法则都要求底数相同才能用。
- 它们符合直觉:$2^2 \cdot 2^3 = 4 \cdot 8 = 32 = 2^5$。
Every step to the right multiplies by the base · 每向右一步都乘以底数
y = a·bˣ
Read the y-value at x = 1, 2, 3. Each is the base times the one before — that is why bˣ⁺¹ = bˣ · b. · 读出 x = 1、2、3 处的 y 值。每一个都是前一个乘以底数——这就是 bˣ⁺¹ = bˣ · b 的原因。
The product rule for exponents says $b^m \cdot b^n =$ · 指数的乘法法则说 $b^m \cdot b^n =$
Multiplying powers of the same base adds the exponents: $b^m \cdot b^n = b^{m+n}$. · 同底数的幂相乘要把指数相加:$b^m \cdot b^n = b^{m+n}$。
The quotient rule says $\dfrac{b^m}{b^n} = b^{m-n}$. · 除法法则说 $\dfrac{b^m}{b^n} = b^{m-n}$。
Dividing powers of the same base subtracts the exponents — the mirror image of the product rule. · 同底数的幂相除要把指数相减——它是乘法法则的镜像。
The power rule
- Raising a power to another power multiplies the exponents: the power rule 幂法则 $(b^m)^n = b^{mn}$.
- So $(2^3)^2 = 2^6 = 64$, matching $8^2 = 64$.
- Each unit step to the right on the graph multiplies the output by the base.
- These rules together explain why exponentials grow the way they do.
幂法则
- 幂的幂,指数相乘:幂法则(power rule)$(b^m)^n = b^{mn}$。
- 所以 $(2^3)^2 = 2^6 = 64$,与 $8^2 = 64$ 一致。
- 在图像上每向右一个单位,输出就乘以底数。
- 这些法则合在一起,解释了指数函数为什么那样增长。
The power rule says $(b^m)^n =$ · 幂法则说 $(b^m)^n =$
Raising a power to a power multiplies the exponents: $(b^m)^n = b^{mn}$. · 幂的幂要把指数相乘:$(b^m)^n = b^{mn}$。
Shifting the exponent
- A shift in the exponent becomes a constant multiplier: $b^{x+k} = b^x \cdot b^k$.
- So $2^{x+3} = 2^x \cdot 8$ — moving the graph left is the same as scaling it up.
- This links transformations of exponentials to the exponent rules.
- A vertical stretch and a horizontal shift can be the same exponential in disguise.
平移指数
- 指数上的平移变成一个常数倍数:$b^{x+k} = b^x \cdot b^k$。
- 所以 $2^{x+3} = 2^x \cdot 8$——把图像左移和把它放大是一回事。
- 这把指数函数的变换与指数法则联系了起来。
- 一次竖直拉伸和一次水平平移,可能是同一个指数函数的伪装。
Using $b^{x+k} = b^x \cdot b^k$, we can rewrite $2^{x+3}$ as $2^x$ times . · 利用 $b^{x+k} = b^x \cdot b^k$,可以把 $2^{x+3}$ 改写成 $2^x$ 乘以。
$2^{x+3} = 2^x \cdot 2^3 = 2^x \cdot 8$ — a shift in the exponent becomes a constant multiplier. · $2^{x+3} = 2^x \cdot 2^3 = 2^x \cdot 8$——指数上的平移变成了一个常数倍数。
Changing the base
- Any exponential can be rewritten with a different base when it helps.
- For example $4^x = (2^2)^x = 2^{2x}$ — same function, base $2$ instead of $4$.
- Rewriting to a common base shows two exponentials are equivalent 等价.
- Choose the base that makes the comparison or calculation easiest.
换底
- 当有帮助时,任何指数函数都可以改写成不同的底数。
- 例如 $4^x = (2^2)^x = 2^{2x}$——同一个函数,底数用 $2$ 而不是 $4$。
- 改写成共同的底数能显示两个指数函数是等价(equivalent)的。
- 选择让比较或计算最容易的那个底数。
Select all · 所有 correct exponent rules. · 选出所有正确的指数法则。
You cannot combine $b^m + b^n$ into one power — the rules apply to multiplying and dividing, not adding. The other three are correct. · 不能把 $b^m + b^n$ 合并成一个幂——这些法则适用于乘除,而不是加法。其余三条正确。
The rules apply to multiplying and dividing powers, never to adding them. $b^m + b^n$ does not simplify to $b^{m+n}$ — that is a classic mistake. You can only combine exponents when the powers are multiplied or divided.
这些法则适用于乘法和除法的幂,绝不适用于加法。$b^m + b^n$ 不能化简为 $b^{m+n}$——这是一个经典错误。只有当幂相乘或相除时,你才能合并指数。
Simplify $\dfrac{2^{x+3}}{2^x}$.
- Quotient rule: subtract the exponents → $2^{(x+3) - x} = 2^3$.
- So the whole expression equals $2^3 = 8$, a constant.
- The $2^x$ cancels because both terms share the base $2$.
化简 $\dfrac{2^{x+3}}{2^x}$。
- 除法法则:指数相减 → $2^{(x+3) - x} = 2^3$。
- 所以整个式子等于 $2^3 = 8$,一个常数。
- $2^x$ 被约掉,因为两项共享底数 $2$。
The exponent rules let you rearrange powers of the same base: the product rule adds exponents, the quotient rule subtracts, and the power rule multiplies. A shift $b^{x+k} = b^x \cdot b^k$ turns into a multiplier, and rewriting to a common base shows two expressions are equivalent.
指数法则让你重排同底数的幂:乘法法则指数相加,除法法则相减,幂法则相乘。平移 $b^{x+k} = b^x \cdot b^k$ 变成一个倍数,而改写成共同底数能显示两个式子是等价的。