Manipulating Exponential Expressions
Introduced| English |
|---|
| quotient rule/ˈkwəʊʃənt ruːl/ |
| base/beɪs/ |
| product rule/ˈprɒdʌkt ruːl/ |
| power rule/ˈpaʊə ruːl/ |
| equivalent/ɪˈkwɪvələnt/ |
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.
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: $2^2 \cdot 2^3 = 4 \cdot 8 = 32 = 2^5$.
Every step to the right multiplies by the base
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.
The product rule for exponents says $b^m \cdot b^n =$
Multiplying powers of the same base adds the exponents: $b^m \cdot b^n = b^{m+n}$.
The quotient rule says $\dfrac{b^m}{b^n} = b^{m-n}$.
Dividing powers of the same base subtracts the exponents — the mirror image of the product rule.
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.
The power rule says $(b^m)^n =$
Raising a power to a power multiplies the exponents: $(b^m)^n = b^{mn}$.
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.
Using $b^{x+k} = b^x \cdot b^k$, we can rewrite $2^{x+3}$ as $2^x$ times ____.
$2^{x+3} = 2^x \cdot 2^3 = 2^x \cdot 8$ — a shift in the exponent becomes a constant multiplier.
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.
Select all correct exponent rules.
You cannot combine $b^m + b^n$ into one power — the rules apply to multiplying and dividing, not adding. The other three are correct.
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.
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 $2$.
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.