Manipulating Exponential Expressions
Introduced| English | Meaning |
|---|---|
| base/beɪs/ | The number raised to a power in an exponential expression. |
| product rule/ˈprɒdʌkt ruːl/ | The differentiation rule for two multiplied functions: the first times the derivative of the second plus the second times the derivative of the first. |
| quotient rule/ˈkwəʊʃənt ruːl/ | The differentiation rule for a ratio: the denominator times the numerator's derivative minus the numerator times the denominator's derivative, divided by the denominator squared. |
| power rule/ˈpaʊə ruːl/ | The differentiation rule stating that the derivative of x raised to n is n multiplied by x raised to (n - 1). |
| equivalent/ɪˈkwɪvələnt/ | Having the same value, meaning or effect in the relationship being considered. |
| root/ruːt/ | A number that gives a specified value when raised to a specified power. |
Two formulas, the same curve
- The expressions $2^{2x-1}$ and $0.5(4)^x$ look different. Yet they give the same output for every real $x$, not just for a few selected inputs.
- Exponent rules explain why. Rewriting a function can reveal a shift, a scale factor or a new base 底数 without changing the function itself.
Products and quotients need a common base
- For $b>0$ and real exponents, the product rule 乘法法则 is $b^m b^n=b^{m+n}$, and the quotient rule 除法法则 is $\dfrac{b^m}{b^n}=b^{m-n}$. The denominator is nonzero because a positive base has positive powers.
- Keep the base the same before combining exponents. For example, $2^2(2^3)=2^5=32$, while $2^2(3^3)$ cannot be combined by adding those exponents over one of the original bases.
Consider a quotient that simplifies for every real $x$:
Subtract the entire denominator exponent; the variable cancels.
The product rule for exponents says $b^m \cdot b^n =$
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Multiplying powers of the same base adds the exponents: $b^m \cdot b^n = b^{m+n}$.
The quotient rule says $\dfrac{b^m}{b^n}=b^{m/n}$ for b>0.
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False. Subtract the exponents: $\dfrac{b^m}{b^n}=b^{m-n}$. The base stays unchanged.
A power of a power multiplies exponents
- The power rule 幂法则 gives $(b^m)^n=b^{mn}$ for a positive base. Thus $(2^3)^2=2^6=64$; the exponent is $3\times2$, not $3+2$.
- This also works for real inputs: $4^x=(2^2)^x=2^{2x}$. The formulas are equivalent 等价 because the identity holds for every real $x$.
The power rule says $(b^m)^n =$
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Raising a power to a power multiplies the exponents: $(b^m)^n = b^{mn}$.
Zero and negative exponents
- A positive base satisfies $b^0=1$. A negative exponent takes a reciprocal: $b^{-n}=1/b^n$, so $2^{-3}=1/8$. It does not put a minus sign in front of the output.
- For a variable exponent, $2^{-(x+1)}=1/2^{x+1}$. At $x=2$, this gives $1/8$, while $-2^{x+1}$ gives $-8$. Parentheses and the position of the minus sign matter.
What is $2^{-3}$?
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$2^{-3}=1/2^3=1/8=0.125$, a positive reciprocal.
Fractional exponents are roots
- For positive $b$ and a positive integer $n$, $b^{1/n}$ is the $n$th root 方根 of $b$. More generally, $b^{m/n}=(\sqrt[n]{b})^m$.
- For example, $16^{3/4}=(\sqrt[4]{16})^3=2^3=8$. Also $9^{x/2}=(9^{1/2})^x=3^x$: a root can become the new base.
What is $16^{3/4}$?
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The fourth root of 16 is 2; cube it to obtain 8.
An input shift becomes a vertical factor
- The product rule gives $b^{x+k}=b^k b^x$. Relative to $b^x$, a positive $k$ shifts the graph left by $k$ units; a negative $k$ shifts it right by $|k|$ units.
- The equivalent vertical factor is $b^k$. Thus $2^{x+3}=8(2)^x$, while $(0.5)^{x+3}=0.125(0.5)^x$. A left shift stretches the first curve vertically but shrinks the second: left does not always mean a larger output.
Using $b^{x+k} = b^x \cdot b^k$, we can rewrite $2^{x+3}$ as $2^x$ times ____.
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$2^{x+3} = 2^x \cdot 2^3 = 2^x \cdot 8$ — a shift in the exponent becomes a constant multiplier.
Which rewrite of $(0.5)^{x+3}$ is correct?
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The multiplier is $(0.5)^3=0.125$. A left shift shrinks this positive decay curve vertically.
An input dilation changes the base
- The power rule gives $b^{cx}=(b^c)^x$. With $c>1$, the horizontal scale factor is $1/c$, so $2^{2x}=4^x$ is a horizontal compression of $2^x$. With $0<c<1$, the graph stretches horizontally.
- A negative $c$ reflects the graph across the vertical axis and scales it horizontally by $1/|c|$: $2^{-x}=(0.5)^x$. For a nonconstant exponential, require $b>0$, $b\ne1$ and $c\ne0$; $c=0$ instead gives the constant 1.
Write $2^{2x}=B^x$. What is B?
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$B=2^2=4$. A horizontal compression changes the base in the equivalent formula.
Combine the two rewrites
- Use the product rule for the added exponent and the power rule for the multiplied exponent. The general identity is $b^{cx+k}=b^k(b^c)^x$.
- For the opening example, the vertical factor is $2^{-1}=0.5$ and the new base is $2^2=4$. Checking sample values helps catch mistakes; the algebra establishes equivalence for all real inputs.
At $x=1$, both give 2; at $x=2$, both give 8. Writing $2x-1=2(x-0.5)$ also shows a horizontal compression followed by a shift right of 0.5 relative to $2^x$.
One function, two equivalent forms
$b^{cx+k}=b^k(b^c)^x$
The default $2^{2x-1}=0.5(4)^x$ shows two equivalent forms. Compare the added exponent k with the input multiplier c, including decay bases and negative c.
Write $2^{2x-1}=A(4)^x$. What is A?
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$A=2^{-1}=0.5$; the added exponent becomes the coefficient.
A sum needs a different argument
- The product rule does not say $b^m+b^n=b^{m+n}$. For example, $2^2+2^3=4+8=12$, while $2^{2+3}=32$. This counterexample disproves the claimed identity.
- You may still factor a sum: $2^x+2^{x+1}=2^x(1+2)=3(2)^x$. The coefficient 3 comes from adding the remaining factors, not from adding the exponents.
Distinguish a product of powers, a power of a power and a sum. Also distinguish $b^{-x}$ from $-b^x$. Each expression has a different operation, even when the symbols look similar.
For positive bases, products add exponents, quotients subtract, powers multiply, negative exponents take reciprocals and fractional exponents give roots. A shift becomes the factor $b^k$; an input dilation becomes the base $b^c$. Preserve the meaning of the whole expression at every step.
For a positive base b and real exponents, select all correct identities.
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You cannot combine $b^m + b^n$ into one power — the rules apply to multiplying and dividing, not adding. The other three are correct.
What is $2^2+2^3$?
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Add the values: 4+8=12. Adding exponents would incorrectly treat this sum as a product.
Justify a product and compare operations
- Calculate $2^5(2^3)$ and explain why its exponents are added. State how the calculation would differ for $(2^5)^3$.
- Explain why rewriting $2^{x+3}$ as $2^x(2^3)$ uses the same product property, even though one expression emphasizes a horizontal shift.
Calculate $2^5(2^3)$ and explain the operation on the exponents. Contrast this with $(2^5)^3$, then explain why $2^{x+3}=2^x(2^3)$ uses the product rule.
Identify the outer operation first. The product rule can combine powers or separate an exponent sum.
What is the product $2^5(2^3)$?
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Same base, so the exponents add.