Radicals and Rational Exponents: Chapter Review

Vocabulary and notation

Index and radicand

In $\sqrt[n]{a}$ the integer $n \ge 2$ is the index and $a$ the radicand; a square root is $n = 2$, with the index left unwritten.

Principal root

What $\sqrt[n]{a}$ names: for $a \ge 0$, the nonnegative real whose $n$-th power is $a$; for $a < 0$, the one (negative) real root when $n$ is odd, and nothing real when $n$ is even.

Like radicals

Two radical terms with the same index and the same radicand, judged only in simplest form. $\sqrt2$ and $\sqrt3$ are unlike; so are $\sqrt2$ and $\sqrt[3]{2}$.

Simplest radical form

No perfect $n$-th power factor left under the radical, no radical in a denominator, and no common factor between the index and the exponents inside.

Radical conjugate $\alpha^{*}$

The partner of $\alpha = a + b\sqrt d$ with the sign of the surd term alone flipped: $\alpha^{*} = a - b\sqrt d$.

Canonical form $p + q\sqrt d$

The unique way to write such a number with $p, q$ rational; two expressions are the same number exactly when both parts match.

Radical equation

An equation with the unknown under a radical or carrying a fractional exponent, so it cannot be freed by adding or dividing.

Extraneous root

A candidate satisfying the equation you produced but not the equation you started with.

Formulas and theorems

• Index parity: when a root is real

  $$\begin{gathered} n \text{ odd:}\ \sqrt[n]{a}\ \text{is real for every real } a \\ n \text{ even:}\ \sqrt[n]{a}\ \text{is real only for } a \ge 0 \end{gathered}$$

  Use when Integer $n \ge 2$, over the reals. An even-index radical is never negative; an odd-index one carries its radicand's sign. With $\sqrt{-1} = i$ it is complex, not meaningless.

• Root of a matching power

  $$\begin{gathered} n \text{ even:}\ \sqrt[n]{a^{n}} = \lvert a \rvert \\ n \text{ odd:}\ \sqrt[n]{a^{n}} = a \end{gathered}$$

  Use when Every real $a$; $\sqrt{x^2} = x$ exactly when $x \ge 0$. Bars go on exactly when the exponent leaving an even root is odd.

  e.g. $\sqrt{x^6} = \lvert x \rvert^3$, while $\sqrt{x^4} = x^2$ and $\sqrt[3]{x^3} = x$ need none.

• Product rule

  $$\sqrt[n]{ab} = \sqrt[n]{a}\,\sqrt[n]{b}$$

  Use when $a \ge 0$ and $b \ge 0$ for even $n$; all real $a, b$ for odd $n$. With BOTH radicands negative under a SQUARE root it fails by a factor of $-1$: convert to $i$ form first, then multiply.

• Quotient rule

  $$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$$

  Use when $a \ge 0$ and $b > 0$ for even $n$; any real $a$ and nonzero real $b$ for odd $n$. Read right to left to collapse an awkward quotient.

  e.g. $\dfrac{\sqrt{98}}{\sqrt2} = \sqrt{49} = 7$.

• Rational exponents

  $$\begin{gathered} a^{1/n} = \sqrt[n]{a} \\ a^{m/n} = \left(\sqrt[n]{a}\right)^{m} = \sqrt[n]{a^{m}} \end{gathered}$$

  Use when $a > 0$, $m$ an integer, $n$ a positive integer. Both routes then agree, whatever fraction is written for the exponent; take the root first for smaller arithmetic. For $a < 0$ the routes disagree and the symbol names nothing, though $(-8)^{1/3} = -2$ may still be written by the odd-root convention.

  e.g. $32^{-2/5} = \dfrac{1}{\left(\sqrt[5]{32}\right)^{2}} = \dfrac14$.

• Exponent laws at rational exponents

  $$\begin{gathered} a^{r}a^{s} = a^{r+s}, \quad \frac{a^{r}}{a^{s}} = a^{r-s} \\ \left(a^{r}\right)^{s} = a^{rs}, \quad (ab)^{r} = a^{r}b^{r} \\ a^{-r} = \frac{1}{a^{r}} \end{gathered}$$

  Use when Bases $a, b > 0$, exponents $r, s$ rational. A negative exponent takes a reciprocal, never a sign. None is guaranteed for a negative base.

• Collecting like radicals

  $$c\sqrt[n]{k} + d\sqrt[n]{k} = (c + d)\sqrt[n]{k}$$

  Use when Same index and same radicand, both terms already in simplest form. The radicand is the unit and never changes as you count more of it.

• Conjugate product

  $$\begin{gathered} (a + b\sqrt d)(a - b\sqrt d) = a^2 - b^2 d \\ (\sqrt m + \sqrt n)(\sqrt m - \sqrt n) = m - n \end{gathered}$$

  Use when $a, b$ rational and $d$ a positive non-square for the first; $m, n \ge 0$ with $m \neq n$ for the second. Dividing by it needs $a^2 \neq b^2 d$, automatic once $\sqrt d$ is irrational and $a, b$ are not both zero.

  e.g. $(3 + 2\sqrt2)(3 - 2\sqrt2) = 9 - 4(2) = 1$.

• Clearing a single radical

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

  Use when $a > 0$. Multiply by whatever power fills the root up to a whole one, less than $a^{n-1}$ when the radicand is itself a power.

  e.g. $\sqrt[3]{9} = 3^{2/3}$ needs only $\sqrt[3]{3}$, so $\dfrac{6}{\sqrt[3]{9}} = 2\sqrt[3]{3}$.

• Cube-root multipliers

  $$\begin{gathered} (x + y)(x^2 - xy + y^2) = x^3 + y^3 \\ (x - y)(x^2 + xy + y^2) = x^3 - y^3 \end{gathered}$$

  Use when Any reals, used with $x$ or $y$ a cube root. For $a + b\sqrt[3]{d}$ the product is $a^3 + b^3 d$, nonzero whenever $\sqrt[3]{d}$ is irrational and the rationals $a, b$ are not both zero.

  e.g. $\left(\sqrt[3]{5} - 1\right)\left(\sqrt[3]{25} + \sqrt[3]{5} + 1\right) = 5 - 1 = 4$.

• What raising both sides does

  $$\begin{gathered} A^2 = B^2 \iff A = B \ \text{or} \ A = -B \\ a^3 = b^3 \iff a = b \end{gathered}$$

  Text description

  Squaring reflects the line x minus 5 in the x-axis, and the square-root curve meets that mirror line at x = 2, the extraneous root, while the true solution x = 9 sits on the original line.

  Use when All four letters real. An even power glues an equation to its sign-flipped twin, so solutions are gained but never lost, and the extraneous ones are the twin's. An odd power is reversible.

• Rationalizing a numerator

  $$\sqrt{n+1} - \sqrt{n} = \frac{1}{\sqrt{n+1} + \sqrt{n}}$$

  Use when $n \ge 0$. Trades a difference of near-equal roots for a division that crude bounds pin tightly.

  e.g. $\sqrt{101} - 10 = \dfrac{1}{\sqrt{101} + 10}$, and $20 < \sqrt{101} + 10 < 20.1$ pins it between $\tfrac{1}{20.1}$ and $\tfrac{1}{20}$.

Problem types, step by step

Exam traps

• Trap Writing $\sqrt{x^2} = x$, which at $x = -3$ reports $-3$ while $\sqrt{(-3)^2} = \sqrt9 = 3$.

  Fix $\sqrt{x^2} = \lvert x \rvert$ for every real $x$, and the two agree exactly when $x \ge 0$.

• Trap Barring every extracted root, turning $\sqrt[3]{x^3}$ into $\lvert x \rvert$ and $\sqrt{x^4}$ into $\lvert x^2 \rvert$.

  Fix Bars belong only where an odd exponent leaves an even root with the variable unrestricted. $\sqrt[3]{x^3} = x$; at $x = -1$ bars would flip its sign.

• Trap Merging two negative radicands, giving $\sqrt{-4}\cdot\sqrt{-9} = \sqrt{36} = 6$.

  Fix The value is $-6$. Convert each radical to $i$ form first: $(2i)(3i) = 6i^2 = -6$.

• Trap Writing $\sqrt a + \sqrt b = \sqrt{a+b}$, so $\sqrt9 + \sqrt{16}$ comes out as $\sqrt{25} = 5$.

  Fix It is $3 + 4 = 7$, and the rule holds only when $a$ or $b$ is $0$. Squaring the left leaves a cross term $2\sqrt{ab}$ the false rule discards.

• Trap Filtering radical-equation candidates by a domain check, so $x = 2$ passes in $\sqrt{x+7} = x-5$ because its radicand is $9$.

  Fix That check belongs to rational equations. The original gives $3 \neq -3$, so $x = 2$ is extraneous despite lying inside the domain.

• Trap Squaring $\sqrt{3x+1} - 2 = 3$ as it stands, or stopping after one squaring when two radicals were present.

  Fix Isolate before raising, or the cross term $-4\sqrt{3x+1}$ frees nothing; then isolate and raise again for each radical that survives.

• Trap Flipping the sign at a cube root: $(1 + \sqrt[3]{2})(1 - \sqrt[3]{2}) = 1 - \sqrt[3]{4}$.

  Fix That swaps one radical for another. A cube root has two irrational powers, so use $1 - \sqrt[3]{2} + \sqrt[3]{4}$, whose product is $3$.

• Trap Leaving the coefficient unsquared, making $(3 + 2\sqrt2)(3 - 2\sqrt2)$ into $9 - 2 = 7$.

  Fix The product is $a^2 - b^2 d = 9 - 4(2) = 1$. The coefficient is squared along with the radical.

• Trap Running the exponent laws over a negative base, as in $(-8)^{2/6} = \left((-8)^2\right)^{1/6} = 2$.

  Fix Reducing the exponent first gives $(-8)^{1/3} = -2$, so no number is named. The laws need $a > 0$; an odd root of a negative may be written but not pushed through them.

• Trap Adding or multiplying indices for a product of different-index radicals, so $\sqrt[3]{2}\cdot\sqrt[6]{2}$ becomes $\sqrt[9]{2}$.

  Fix The EXPONENTS add, not the indices: $2^{1/3}\cdot2^{1/6} = 2^{1/2} = \sqrt2$.

• Trap Stopping the moment a denominator turns rational, leaving $\dfrac{4(5+\sqrt7)}{18}$.

  Fix Numerator and denominator still share a factor of $2$, so the finished answer is $\dfrac{10 + 2\sqrt7}{9}$.

• Trap Rejecting a negative candidate on sight, or refusing a negative right-hand side at an odd index.

  Fix A negative $x$ is fine while every radicand stays nonnegative, and $x = -1$ solves $\sqrt{2x+3} - \sqrt{x+1} = 1$. Only an even root set equal to a negative is impossible.