Rational exponents and logarithm laws Notes | Mathematics AA HL

Meaning of rational and fractional exponents

A rational exponent is an exponent that can be written as a fraction $\frac{m}{n}$ , where $m\in\mathbb Z$ , $n\in\mathbb N$ , and $n>0$ . The denominator tells us which root to take and the numerator tells us which power to apply.

For a positive real base $a$ , the definition is $a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$ . Both forms are equivalent, but taking the root first often keeps numbers smaller and makes mental calculation easier.

A negative rational exponent means reciprocal, exactly as it does for integer exponents: $a^{\frac{-m}{n}}=\frac{1}{a^{\frac{m}{n}}}$ . It does not make the value negative.

Real-number domain conditions matter. An even root requires a non-negative radicand, while an odd root can accept a negative radicand. For most exponent-law manipulations at AA SL, assuming a positive base avoids ambiguity and guarantees that every expression is real.

Key terms

Rational exponent: An exponent of the form $\frac{m}{n}$ , where $m$ and $n$ are integers and $n>0$ .
Index: The root number $n$ in $\sqrt[n]{a}$ ; it becomes the denominator of the rational exponent.
Radicand: The expression inside a radical sign.
Reciprocal: The multiplicative inverse $\frac{1}{x}$ ; a negative exponent creates a reciprocal.

Worked examples

Interpret a fractional exponent

Evaluate $27^{\frac{2}{3}}$ .

1The denominator 3 means take a cube root.
2 $\sqrt[3]{27}=3$ .
3The numerator 2 means square the result: $3^2=9$ .

Answer: $27^{\frac{2}{3}}=9$ .

Use a negative rational exponent

Evaluate $16^{\frac{-3}{4}}$ .

1The negative sign means take the reciprocal.
2 $16^{\frac{1}{4}}=2$ , so $16^{\frac{3}{4}}=2^3=8$ .
3Take the reciprocal: $16^{\frac{-3}{4}}=\frac{1}{8}$ .

Answer: $\frac18$ .

Check the real domain

Compare $(-32)^{\frac{2}{5}}$ and $(-16)^{\frac{1}{4}}$ over the real numbers.

1A fifth root is an odd root, so $\sqrt[5]{-32}=-2$ and $(-2)^2=4$ .
2A fourth root is an even root, and no real number has fourth power $-16$ .

Answer: $(-32)^{\frac{2}{5}}=4$ , while $(-16)^{\frac{1}{4}}$ is not real.

Rules and formulas

Rational exponent definition

a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m

Negative rational exponent

a^{\frac{-m}{n}}=\frac{1}{a^{\frac{m}{n}}}

Special root form

a^{\frac{1}{n}}=\sqrt[n]{a}

Converting between radicals and rational exponents

Radical notation and rational-exponent notation express the same operation. Converting to exponent notation is especially useful when multiplying, dividing, differentiating later in the course, or entering expressions into a GDC.

For $x>0$ , integer $m$ and positive integer $n$ , use $\sqrt[n]{x^m}=x^{\frac{m}{n}}$ . A coefficient outside the radical stays outside. Splitting a radical of a product into radicals of its factors is safe when those factors are positive; check domains separately for zero or negative factors.

When simplifying even roots of variable squares, remember absolute value: $\sqrt{x^2}=|x|$ . If a question states $x>0$ , then $|x|=x$ ; without that condition, removing the absolute value is not justified.

Key terms

Radical form: Notation using a root symbol, such as $\sqrt[3]{x^2}$ .
Exponent form: Equivalent power notation, such as $x^{\frac{2}{3}}$ .
Absolute value: The non-negative magnitude $|x|$ , required because the principal square root is never negative.

Worked examples

Radical to exponent form

Write $\sqrt[5]{x^3}$ using a rational exponent.

1The index 5 becomes the denominator.
2The power 3 becomes the numerator.

Answer: $\sqrt[5]{x^3}=x^{\frac{3}{5}}$ .

Exponent to radical form

Write $y^{\frac{7}{4}}$ in radical form.

1The denominator 4 gives a fourth root.
2The numerator 7 gives the power.

Answer: $y^{\frac{7}{4}}=\sqrt[4]{y^7}=(\sqrt[4]{y})^7$ .

Simplify a radical exactly

Simplify $\sqrt[3]{54x^7}$ .

1Factor perfect cubes: $54x^7=27x^6(2x)$ .
2Take the cube root of $27x^6$ : $\sqrt[3]{27x^6}=3x^2$ .
3Leave the remaining factor inside the cube root.

Answer: $3x^2\sqrt[3]{2x}$ .

Rules and formulas

Radical–exponent conversion

\sqrt[n]{x^m}=x^{\frac{m}{n}},\qquad x>0,\ m\in\mathbb Z,\ n\in\mathbb Z_{>0}

Principal square root

\sqrt{x^2}=|x|

Odd root of a power

\sqrt[2k+1]{x^{2k+1}}=x

Laws of exponents with rational powers

For positive bases, the usual exponent laws hold for rational exponents: combine powers of the same base by adding or subtracting exponents, and multiply exponents for a power of a power. For negative bases, merely having real, defined expressions is not enough to guarantee every exponent law; check the domain and principal-root convention separately.

When multiplying powers with the same base, add exponents. When dividing, subtract exponents. When raising a power to another power, multiply exponents.

A power distributes over a product or quotient, but it does not distribute over a sum: $(x+y)^r\ne x^r+y^r$ in general.

Zero deserves special care: $0^r$ is defined for positive $r$ , but $0^0$ and $0^r$ for negative $r$ are undefined.

Key terms

Product law: Same-base powers multiply by adding their exponents.
Quotient law: Same-base powers divide by subtracting their exponents.
Power of a power: Raising $a^r$ to $s$ multiplies the exponents.
Domain restriction: A condition required to keep every algebraic expression defined in the chosen number system.

Worked examples

Multiply rational powers

Simplify $x^{\frac{2}{3}}x^{\frac{5}{6}}$ for $x>0$ .

1Use a common denominator: $\frac{2}{3}=\frac{4}{6}$ .
2Add exponents: $\frac{4}{6}+\frac{5}{6}=\frac{9}{6}=\frac{3}{2}$ .

Answer: $x^{\frac{3}{2}}$ .

Divide rational powers

Simplify $\frac{a^{\frac{7}{4}}}{a^{\frac{1}{2}}}$ for $a>0$ .

1Rewrite $\frac{1}{2}$ as $\frac{2}{4}$ .
2Subtract exponents: $\frac{7}{4}-\frac{2}{4}=\frac{5}{4}$ .

Answer: $a^{\frac{5}{4}}$ .

Power of a power

Simplify $(p^{\frac{3}{5}})^{\frac{10}{9}}$ for $p>0$ .

1Multiply the exponents: $(\frac{3}{5})(\frac{10}{9})=\frac{30}{45}=\frac{2}{3}$ .

Answer: $p^{\frac{2}{3}}$ .

Rules and formulas

Product law

a^r a^s=a^{r+s}

Quotient law

\frac{a^r}{a^s}=a^{r-s}

Power of a power

(a^r)^s=a^{rs}

Power of a product

(ab)^r=a^r b^r

Power of a quotient

\left(\frac ab\right)^r=\frac{a^r}{b^r}

Simplifying expressions involving rational exponents

A strong simplification keeps values exact, combines like bases, removes negative exponents when requested, and states domain conditions when they matter.

Work in a consistent notation. Convert radicals to powers before using exponent laws, then convert back to radical form only if it makes the final answer clearer.

Factor coefficients into perfect powers. For example, recognizing $81=3^4$ makes fourth roots immediate.

Key terms

Exact value: A value written without premature decimal approximation, such as $3\sqrt2$ or $2^{\frac{5}{3}}$ .
Positive exponent form: A final form with negative exponents rewritten as reciprocals.
Like bases: Powers sharing the same base, which can be combined using exponent laws.

Worked examples

Combine several powers

Simplify $\frac{x^{\frac{3}{2}}x^{\frac{-1}{4}}}{x^{\frac{1}{2}}}$ for $x>0$ .

1Add numerator exponents and subtract the denominator exponent.
2 $\frac{3}{2}-\frac{1}{4}-\frac{1}{2}=\frac{6}{4}-\frac{1}{4}-\frac{2}{4}=\frac{3}{4}$ .

Answer: $x^{\frac{3}{4}}$ .

Simplify a numerical expression

Evaluate $81^{\frac{3}{4}}\cdot 8^{\frac{1}{3}}$ .

1 $81^{\frac{1}{4}}=3$ , so $81^{\frac{3}{4}}=3^3=27$ .
2 $8^{\frac{1}{3}}=2$ .
3Multiply: $27\cdot2=54$ .

Answer: $54$ .

Remove a negative exponent

Simplify $(16x^{-8})^{\frac{3}{4}}$ for $x>0$ .

1Apply the power to each factor.
2 $16^{\frac{3}{4}}=8$ and $(x^{-8})^{\frac{3}{4}}=x^{-6}$ .
3Rewrite with a positive exponent.

Answer: $\frac{8}{x^6}$ .

Rules and formulas

Negative exponent conversion

x^{-r}=\frac{1}{x^r}

Common-denominator strategy

x^{\frac{p}{q}}x^{\frac{r}{s}}=x^{\frac{(ps+rq)}{(qs)}}

Exact before decimal

\sqrt[n]{a^m}=a^{\frac{m}{n}}

Review of logarithms and their domains

A logarithm answers an exponent question. The statement $\log_a x=y$ means that $a^y=x$ . Exponential form and logarithmic form contain the same information.

For real logarithms, the base must satisfy $a>0$ and $a\ne1$ , while the argument must satisfy $x>0$ . These restrictions must be checked before and after solving equations.

The common logarithm $\log x$ usually means base 10, while $\ln x$ means base $e$ , where $e\approx2.71828$ .

Two anchor facts follow directly from exponent rules: $\log_a1=0$ because $a^0=1$ , and $\log_a a=1$ because $a^1=a$ .

Key terms

Logarithm: The exponent to which a base must be raised to produce a given positive argument.
Base: The positive number $a\ne1$ in $\log_a x$ .
Argument: The positive expression inside a logarithm.
Natural logarithm: The logarithm with base $e$ , written $\ln$ .

Worked examples

Convert to logarithmic form

Write $2^5=32$ in logarithmic form.

1The base is 2, the exponent is 5, and the result is 32.

Answer: $\log_2 32=5$ .

Convert to exponential form

Write $\log_3(\frac{1}{27})=-3$ in exponential form.

1Keep base 3, use exponent $-3$ , and set the result equal to $\frac{1}{27}$ .

Answer: $3^{-3}=\frac{1}{27}$ .

Find the domain

Find the domain of $f(x)=\ln(2x-6)$ .

1The logarithm argument must be positive.
2Solve $2x-6>0$ .
3Therefore $x>3$ .

Answer: The domain is $(3,\infty)$ .

Rules and formulas

Logarithm definition

\log_a x=y\quad\Longleftrightarrow\quad a^y=x

Logarithm of one

\log_a1=0

Logarithm of its base

\log_a a=1

Inverse relationships

a^{\log_a x}=x,\qquad \log_a(a^x)=x

Product, quotient, and power laws of logarithms

Logarithm laws convert multiplication into addition, division into subtraction, and powers into coefficients. They follow from the corresponding exponent laws.

The product law applies to a product inside one logarithm. The quotient law applies to a quotient. The power law moves an exponent on the entire argument to the front.

There is no logarithm law for a sum or difference. Expressions such as $\log_a(x+y)$ and $\log_a(x-y)$ generally cannot be split.

Every separate logarithm created by expansion must have a positive argument. When variables appear, use the domain conditions supplied in the question or state appropriate restrictions.

Key terms

Product law: A logarithm of a product becomes the sum of logarithms.
Quotient law: A logarithm of a quotient becomes the difference of logarithms.
Power law: An exponent on a logarithm argument becomes a multiplier.
Identity: An equality true for every value in the common domain of both sides.

Worked examples

Apply all three laws

Expand $\log_2\left(\frac{8x^3}{\sqrt y}\right)$ , where $x,y>0$ .

1Use the quotient law: $\log_2(8x^3)-\log_2(y^{\frac{1}{2}})$ .
2Use product and power laws: $\log_2 8+3\log_2x-\frac12\log_2y$ .
3Since $\log_2 8=3$ , simplify.

Answer: $3+3\log_2x-\frac12\log_2y$ .

Use the power law

Simplify $\log_5 125^x$ .

1Apply the power law: $x\log_5 125$ .
2Since $125=5^3$ , $\log_5 125=3$ .

Answer: $3x$ .

Recognize an invalid split

Is $\ln(x+4)=\ln x+\ln4$ an identity?

1The product law applies to $\ln(4x)$ , not $\ln(x+4)$ .
2Test $x=1$ : $\ln5\ne\ln1+\ln4=\ln4$ .

Answer: No. Logarithms do not split across addition.

Rules and formulas

Product law

\log_a(xy)=\log_a x+\log_a y

Quotient law

\log_a\left(\frac{x}{y}\right)=\log_a x-\log_a y

Power law

\log_a(x^r)=r\log_a x

Reciprocal result

\log_a\left(\frac1x\right)=-\log_a x

Expanding, condensing, and changing the base

Expanding rewrites one logarithm as several simpler terms. Condensing reverses the process and combines several logarithms into one. Coefficients must become exponents before product or quotient laws are used.

The change-of-base formula lets a GDC evaluate logarithms in any valid base using $\log$ or $\ln$ . Both numerator and denominator must use the same new base.

Calculator answers are approximations unless the logarithm simplifies exactly. Retain full precision internally and round only at the final step.

Key terms

Expand: Rewrite a logarithm of products, quotients, or powers as a sum or difference.
Condense: Combine logarithmic terms into a single logarithm.
Change of base: Rewrite a logarithm using another valid base, usually $10$ or $e$ , for evaluation.
GDC: A graphical display calculator used to evaluate, graph, and solve expressions numerically.

Worked examples

Condense a logarithmic expression

Condense $2\ln x-\frac12\ln y+\ln3$ , where $x,y>0$ .

1Use the power law backwards: $\ln(x^2)-\ln(y^{\frac{1}{2}})+\ln3$ .
2Combine addition as multiplication and subtraction as division.

Answer: $\ln\left(\frac{3x^2}{\sqrt y}\right)$ .

Evaluate by change of base

Evaluate $\log_7 20$ to three significant figures.

1Enter $\frac{\ln20}{\ln7}$ or $\frac{\log20}{\log7}$ .
2The calculator gives approximately $1.53950184956$ .
3Round to three significant figures.

Answer: $\log_7 20\approx1.54$ .

Expand before substituting

Given $\log_a2=p$ and $\log_a3=q$ , express $\log_a72$ in terms of $p,q$ .

1Factor $72=2^3\cdot3^2$ .
2Apply product and power laws.
3 $\log_a72=3\log_a2+2\log_a3$ .

Answer: $3p+2q$ .

Rules and formulas

Change of base

\log_a x=\frac{\log_b x}{\log_b a}

Calculator forms

\log_a x=\frac{\ln x}{\ln a}=\frac{\log x}{\log a}

Condensing pattern

r\log_a x-s\log_a y=\log_a\left(\frac{x^r}{y^s}\right)

Solving exponential and logarithmic equations

For exponential equations, first try to express both sides with the same base. If that is impossible, isolate the exponential expression and take logarithms.

For logarithmic equations, state domain restrictions, combine logarithms when useful, convert to exponential form, solve the resulting equation, and check every candidate in the original equation.

Taking logarithms is valid only after the isolated exponential side is positive. This is automatically true for $a^{f(x)}$ when $a>0$ , but coefficients and other terms still need careful algebra.

A GDC can solve equations numerically using a solver or graph intersections. An IB response should still show the equation entered, the chosen interval or graph, and suitable rounding.

Key terms

Extraneous solution: A value produced during algebra that does not satisfy the original equation or its domain.
Same-base method: Rewrite both sides with one base and equate exponents.
Logarithmic method: Take logarithms after isolating an exponential expression.
Intersection method: Graph each side as a function and read the coordinates where the graphs meet.

Worked examples

Solve using a common base

Solve $4^x=8^{x-1}$ .

1Rewrite using base 2: $2^{2x}=2^{3x-3}$ .
2Equate exponents: $2x=3x-3$ .
3Solve to obtain $x=3$ .

Answer: $x=3$ .

Solve by taking logarithms

Solve $3^{2x-1}=20$ .

1Take natural logarithms: $(2x-1)\ln3=\ln20$ .
2Rearrange: $2x-1=\frac{\ln20}{\ln3}$ .
3Therefore $x=\frac12\left(1+\frac{\ln20}{\ln3}\right)$ .

Answer: $x\approx1.86$ to three significant figures.

Solve and reject an invalid root

Solve $\log_2x+\log_2(x-2)=3$ .

1Domain: $x>2$ .
2Combine: $\log_2[x(x-2)]=3$ .
3Convert: $x(x-2)=8$ , so $x^2-2x-8=0$ .
4The candidates are $x=4$ and $x=-2$ ; reject $-2$ because it violates the domain.

Answer: $x=4$ .

Solve a growth equation

Solve $2e^{0.4t}=15$ .

1Isolate the exponential: $e^{0.4t}=7.5$ .
2Take natural logarithms: $0.4t=\ln7.5$ .
3Divide by $0.4$ .

Answer: $t=\frac{\ln7.5}{0.4}\approx5.04$ .

Rules and formulas

Same-base principle

a^u=a^v\quad\Longrightarrow\quad u=v

Logarithmic solution step

a^{f(x)}=k\quad\Longrightarrow\quad f(x)=\frac{\ln k}{\ln a}

Log equality principle

\log_a u=\log_a v\quad\Longrightarrow\quad u=v

IB-style practice with worked solutions

The following questions combine exact simplification, logarithm laws, equations, domains, and calculator technique. Try each question before reading its solution.

In an IB solution, show exact algebra first. If a decimal is requested, give the exact logarithmic form before the rounded value and state the required accuracy.

Key terms

Method mark: Credit awarded for a valid mathematical process even when a later arithmetic error occurs.
Exact form: A form involving fractions, radicals, or logarithms without decimal rounding.
Verification: Substitution or domain checking used to confirm that a candidate is valid.

Worked examples

Practice 1 — rational exponent

Evaluate $64^{\frac{2}{3}}$ .

1 $\sqrt[3]{64}=4$ .
2Square: $4^2=16$ .

Answer: $16$ .

Practice 2 — negative fractional power

Evaluate $81^{\frac{-3}{4}}$ .

1 $81^{\frac{1}{4}}=3$ , so $81^{\frac{3}{4}}=27$ .
2Use the reciprocal for the negative exponent.

Answer: $\frac{1}{27}$ .

Practice 3 — convert notation

Write $\sqrt[5]{x^3}$ using an exponent.

1The root index becomes the denominator and the power becomes the numerator.

Answer: $x^{\frac{3}{5}}$ .

Practice 4 — simplify powers

Simplify $\frac{x^{\frac{3}{2}}x^{\frac{-1}{4}}}{x^{\frac{1}{2}}}$ , where $x>0$ .

1Combine exponents: $\frac{3}{2}-\frac{1}{4}-\frac{1}{2}$ .
2Using denominator 4 gives $\frac{6}{4}-\frac{1}{4}-\frac{2}{4}=\frac{3}{4}$ .

Answer: $x^{\frac{3}{4}}$ .

Practice 5 — expand logarithms

Expand $\log_5\left(\frac{25x^4}{y^2}\right)$ , where $x,y>0$ .

1Use quotient, product, and power laws.
2 $\log_525+4\log_5x-2\log_5y$ .
3Since $25=5^2$ , $\log_525=2$ .

Answer: $2+4\log_5x-2\log_5y$ .

Practice 6 — condense logarithms

Condense $\frac12\ln a+2\ln b-\ln c$ , where $a,b,c>0$ .

1Move coefficients to powers: $\ln(a^{\frac{1}{2}})+\ln(b^2)-\ln c$ .
2Combine using product and quotient laws.

Answer: $\ln\left(\frac{\sqrt a\,b^2}{c}\right)$ .

Practice 7 — change of base

Evaluate $\log_7 20$ to three significant figures.

1Use $\frac{\ln20}{\ln7}$ .
2The calculator value is approximately $1.53950184956$ .

Answer: $1.54$ .

Practice 8 — common base equation

Solve $9^{x-1}=27$ .

1Rewrite as $3^{2x-2}=3^3$ .
2Equate exponents: $2x-2=3$ .

Answer: $x=\frac{5}{2}$ .

Practice 9 — logarithmic quotient

Solve $\ln(x+2)-\ln x=\ln3$ .

1Domain: $x>0$ .
2Combine: $\ln\left(\frac{x+2}{x}\right)=\ln3$ .
3Thus $\frac{(x+2)}{x}=3$ , so $x+2=3x$ .

Answer: $x=1$ .

Practice 10 — domain rejection

Solve $\log_2(x-1)+\log_2(x+1)=3$ .

1Domain: $x>1$ .
2Combine and convert: $(x-1)(x+1)=2^3$ .
3So $x^2-1=8$ , giving $x=\pm3$ .
4Reject $-3$ because it violates $x>1$ .

Answer: $x=3$ .

Practice 11 — exponential modelling

A quantity satisfies $P=350(1.06)^t$ . Find $t$ when $P=500$ .

1Set $350(1.06)^t=500$ , so $(1.06)^t=\frac{10}{7}$ .
2Take logs: $t\ln1.06=\ln(\frac{10}{7})$ .
3Use full GDC precision.

Answer: $t=\frac{\ln(\frac{10}{7})}{\ln1.06}\approx6.12$ .

Rules and formulas

Fractional exponent

a^{\frac{m}{n}}=\sqrt[n]{a^m}

Three log laws

\log_a(xy)=\log_ax+\log_ay,\quad \log_a(\frac{x}{y})=\log_ax-\log_ay,\quad \log_a(x^r)=r\log_ax

Change of base

\log_a x=\frac{\ln x}{\ln a}

Topic summary

For $a>0$ , integer $m$ and positive integer $n$ : $a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$ ; the denominator is the root and the numerator is the power.
A negative rational exponent creates a reciprocal: $a^{\frac{-m}{n}}=\frac{1}{a^{\frac{m}{n}}}$ .
Even roots need non-negative radicands in the real numbers; odd roots may accept negative radicands.
For positive bases, rational powers follow the usual product, quotient, and power-of-a-power laws.
Remember $\sqrt{x^2}=|x|$ , not automatically $x$ .
$\log_a x=y$ means $a^y=x$ , with $a>0$ , $a\ne1$ , and $x>0$ .
Product becomes addition, quotient becomes subtraction, and a power becomes a coefficient in logarithms.
There is no general sum law equating $\log(x+y)$ with $\log x+\log y$ ; these expressions agree only for some special inputs.
Change base using $\log_a x=\frac{\ln x}{\ln a}$ , keeping the same base in numerator and denominator.
Use a common base for exponential equations when possible; otherwise isolate the power and take logs.
For logarithmic equations, state domain restrictions and check all candidates in the original equation.
Keep exact values and full GDC precision until the final requested rounding.