Math 2–3 Honors
Integrated Math 2–3 Honors · Lesson Review · September 8, 2026

Radical Operations and Rationalizing Denominators

Like radicals combine like variables. Conjugates turn a radical denominator into a difference of squares.

29-minute edited lesson11 chaptersCaptions + transcript

01 · Watch

Lesson video

VTT captionsSRT captions
0:0001Radical Operations and Rationalizing Denominators0:0502Combining Like Radicals1:3803Simplify Before You Combine2:5104Distribute Through Radical Expressions4:2905A Multi-Step Radical Product5:2306Multiplying Two Binomials10:0307Conjugates and Difference of Squares15:3508Rationalizing a One-Term Denominator20:3109Rationalizing with a Conjugate24:2010A Larger Conjugate Example28:0911Lesson Summary

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02 · Understand

Learning targets

03 · Connect

Three structures to recognize

Correctly typeset math keeps coefficients, radicals, fractions, and grouping symbols visually distinct.

01Like radicals
an+bn=(a+b)na\sqrt n+b\sqrt n=(a+b)\sqrt nThe radical part must match before the coefficients can be combined.
02Difference of squares
(a+b)(a−b)=a2−b2(a+b)(a-b)=a^2-b^2The middle terms cancel when the two binomials differ only by their sign.
03Rationalizing a denominator
23+5⋅3−53−5=3−52\frac{2}{3+\sqrt5}\cdot\frac{3-\sqrt5}{3-\sqrt5}=\frac{3-\sqrt5}{2}Multiplying by a form of one removes the radical from the denominator.

04 · Apply

Worked examples

Follow the same structure used in the recorded board work.

Combine

Simplify first, then collect like radicals

8+50=22+52=72\sqrt8+\sqrt{50}=2\sqrt2+5\sqrt2=7\sqrt2

Both radicals become multiples of 23+43=632\sqrt3+4\sqrt3=6\sqrt3 before their coefficients are combined.

32+45−62+75=−32+1153\sqrt2+4\sqrt5-6\sqrt2+7\sqrt5=-3\sqrt2+11\sqrt5

Distribute

Multiply every term

3(23+6)=6+32\sqrt3\left(2\sqrt3+\sqrt6\right)=6+3\sqrt2

25(30+38)=106+12102\sqrt5\left(\sqrt{30}+3\sqrt8\right)=10\sqrt6+12\sqrt{10}

Simplify each radical product before combining any like terms.

Binomials

Track all four products and their signs

(2+3)(5−4)=10+35−42−12(\sqrt2+3)(\sqrt5-4)=\sqrt{10}+3\sqrt5-4\sqrt2-12

(5−3)(10+2)=52−310+25−6(\sqrt5-3)(\sqrt{10}+2)=5\sqrt2-3\sqrt{10}+2\sqrt5-6

Conjugates

Switch the middle sign

(2+4)(2−4)=2−16=−14(\sqrt2+4)(\sqrt2-4)=2-16=-14

(3−27)(3+27)=9−28=−19(3-2\sqrt7)(3+2\sqrt7)=9-28=-19

The opposite-sign middle terms cancel, leaving a difference of squares.

One-term denominator

Multiply by the missing radical factor

622⋅22=32\frac{\sqrt6}{2\sqrt2}\cdot\frac{\sqrt2}{\sqrt2}=\frac{\sqrt3}{2}

153⋅33=5\frac{\sqrt{15}}{\sqrt3}\cdot\frac{\sqrt3}{\sqrt3}=\sqrt5

21035⋅55=223\frac{2\sqrt{10}}{3\sqrt5}\cdot\frac{\sqrt5}{\sqrt5}=\frac{2\sqrt2}{3}

Binomial denominator

Multiply by the denominator’s conjugate

23+5⋅3−53−5=3−52\frac{2}{3+\sqrt5}\cdot\frac{3-\sqrt5}{3-\sqrt5}=\frac{3-\sqrt5}{2}

1+23−7⋅3+73+7=−3+6+7+144\frac{1+\sqrt2}{\sqrt3-\sqrt7}\cdot\frac{\sqrt3+\sqrt7}{\sqrt3+\sqrt7}=-\frac{\sqrt3+\sqrt6+\sqrt7+\sqrt{14}}{4}

Distribute in the numerator and use difference of squares in the denominator, then reduce.

05 · Avoid

Common mistakes

Combining unlike radicals.

Simplify first and combine only when the radical parts match.

Missing a distributed product.

Every term in one factor must multiply every term in the other.

Losing a negative sign.

Write all four binomial products before simplifying.

Using the same sign.

A conjugate switches the middle sign: a+b ⟷ a−ba+b\ \longleftrightarrow\ a-b.

Changing only the denominator.

Multiply the entire fraction by the same nonzero expression over itself.

Stopping too early.

Simplify radicals and reduce common numerical factors in the final answer.

06 · Check yourself

Quick check

Try each prompt before opening the worked answer.

1Combine the like radicals in 23+43=632\sqrt3+4\sqrt3=6\sqrt3.

23+43=632\sqrt3+4\sqrt3=6\sqrt3. Add the coefficients and keep the common radical.

2Simplify and combine 8+50=22+52=72\sqrt8+\sqrt{50}=2\sqrt2+5\sqrt2=7\sqrt2.

8+50=22+52=72\sqrt8+\sqrt{50}=2\sqrt2+5\sqrt2=7\sqrt2. Extract the perfect-square factors first.

3What is the conjugate pattern?

(a+b)(a−b)=a2−b2(a+b)(a-b)=a^2-b^2. The middle terms cancel.

4Rationalize 23+5⋅3−53−5=3−52\frac{2}{3+\sqrt5}\cdot\frac{3-\sqrt5}{3-\sqrt5}=\frac{3-\sqrt5}{2}.

23+5⋅3−53−5=3−52\frac{2}{3+\sqrt5}\cdot\frac{3-\sqrt5}{3-\sqrt5}=\frac{3-\sqrt5}{2}. Multiply by the conjugate of the denominator.

07 · Revisit

Transcript and captions

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