The Timpson Lyceum · Intermediate Algebra (2023)

Lecture 17 (Intermediate Algebra) Radicals and Exponents

Radicals and Exponents – Lecture 17

Overview

This lecture pulls together the major skills for working with radicals and exponents. It begins with rationalizing denominators — clearing radicals out of the bottom of a fraction, both for single-term denominators and for two-term denominators using a conjugate. It then connects radical notation to rational (fractional) exponents, reviews the laws of exponents, and uses them to simplify expressions and to convert between roots. Finally it introduces the imaginary unit $i$ and the arithmetic of complex numbers, including rationalizing imaginary denominators with the complex conjugate. These techniques are the standard toolkit for simplifying any expression involving roots, fractional powers, or square roots of negative numbers.

Key Concepts

Square roots and $n$th roots. The square root of a number is the value that, multiplied by itself, gives that number: $\sqrt{n}\cdot\sqrt{n}=n$. More generally a root times itself the right number of times returns the radicand:

Simplifying a radical. Factor the radicand to expose a perfect square (or perfect $n$th power) and pull it out: $\sqrt{18}=\sqrt{9\cdot 2}=3\sqrt{2}$. For variables, halve the exponent that comes out: $\sqrt{x^{5}}=\sqrt{x^{4}\cdot x}=x^{2}\sqrt{x}$.

Rationalizing a single-term denominator. To remove $\sqrt{a}$ from a denominator, multiply the fraction by $\dfrac{\sqrt{a}}{\sqrt{a}}$ (this equals $1$, so the value is unchanged) because $\sqrt{a}\cdot\sqrt{a}=a$ has no radical.

Rationalizing a two-term denominator (the conjugate). When the denominator is a sum or difference such as $\sqrt{a}-b$, multiplying by $\sqrt{a}$ alone fails. Instead multiply top and bottom by the conjugate — the same two terms with the opposite sign between them — using the difference of squares: $(a-b)(a+b)=a^{2}-b^{2}.$ Squaring both terms eliminates any square root. The conjugate of $5+\sqrt{3}$ is $5-\sqrt{3}$, and vice versa.

Rational (fractional) exponents. A fractional exponent is a root: the denominator is the index (outside the radical) and the numerator is the power (inside): $a^{\,n/m}=\sqrt[m]{a^{\,n}}=\left(\sqrt[m]{a}\right)^{n}.$ When evaluating, take the root first to keep numbers small (e.g. $27^{4/3}=(\sqrt[3]{27})^{4}=3^{4}=81$).

Laws of exponents (used throughout):

• $a^{m}\cdot a^{n}=a^{m+n}$
• $\dfrac{a^{m}}{a^{n}}=a^{m-n}$
• $\left(a^{m}\right)^{n}=a^{mn}$
• $(ab)^{m}=a^{m}b^{m}$
• $\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}=\left(\dfrac{b}{a}\right)^{-m}$
• $a^{-m}=\dfrac{1}{a^{m}}$ (a negative exponent moves the factor across the fraction bar)
• $a^{0}=1$

Converting and combining radicals via exponents. Rewrite a root as a fractional power, reduce or combine the exponents, then convert back. The simplification $\sqrt[8]{x^{6}y^{2}}=\sqrt[4]{x^{3}y}$ comes from dividing each exponent fraction by a common factor of $2$. To multiply or divide radicals with different indices, rewrite with a common denominator in the exponents (a common index) first, then combine.

The imaginary unit. Define $i=\sqrt{-1}$, so $i^{2}=-1$. Powers of $i$ cycle with period $4$: $i^{0}=1,\quad i^{1}=i,\quad i^{2}=-1,\quad i^{3}=-i,\quad i^{4}=1.$ For a large power, reduce the exponent to the nearest multiple of $4$: $i^{97}=i^{96}\cdot i = 1\cdot i = i$.

Complex numbers. A complex number $a+bi$ has a real part $a$ and an imaginary part $b$. Add by combining like parts; multiply by distributing (FOIL) and replacing $i^{2}$ with $-1$. To clear an imaginary denominator, multiply by $i$ (if it is a single $i$) or by the complex conjugate $a-bi$ (if it is two terms), again using $(a+b)(a-b)=a^{2}-b^{2}$.

Square roots of negatives. $\sqrt{-c}=\sqrt{c}\,\sqrt{-1}=i\sqrt{c}$; for example $\sqrt{-16}=4i$.

Worked Example

Rationalize a two-term denominator using the conjugate: simplify $\dfrac{15}{\sqrt{5}+\sqrt{3}}$.

Multiply numerator and denominator by the conjugate $\sqrt{5}-\sqrt{3}$, then use $(a+b)(a-b)=a^{2}-b^{2}$ to clear the radicals from the bottom:

Multiply complex numbers: simplify $(2-4i)(3+5i)$.

Distribute every term, then replace $i^{2}$ with $-1$ and combine like parts:

Summary

• To rationalize a single-term denominator, multiply top and bottom by that radical; to rationalize a two-term denominator, multiply by its conjugate and use $(a+b)(a-b)=a^{2}-b^{2}$.
• A fractional exponent is a root: $a^{n/m}=\sqrt[m]{a^{n}}=(\sqrt[m]{a})^{n}$; take the root first when evaluating.
• The laws of exponents (add when multiplying, subtract when dividing, multiply when raising a power to a power, and $a^{-m}=\tfrac{1}{a^{m}}$, $a^{0}=1$) drive every simplification.
• Convert radicals to fractional powers to reduce indices and to combine roots of different indices over a common denominator.
• $i=\sqrt{-1}$ and $i^{2}=-1$; powers of $i$ repeat every $4$, so reduce large exponents to the nearest multiple of $4$.
• Add complex numbers part by part, multiply by distributing and substituting $i^{2}=-1$, and clear imaginary denominators with $i$ or with the complex conjugate.