A. Characteristics of Functions · Quadratic Functions

Exact Answers: Simplifying and Operating with Radicals

A3.2About 70 minutes

Learning goals

I can explain and use the property √(ab) = √a × √b.
I can write a radical in simplest form and change between mixed and entire radicals.
I can add, subtract and multiply radical expressions, including binomial products.
I can simplify expressions like (6 + √12)/2 that come from the quadratic formula.

Words to know

radical
An expression with a root sign, such as √8.

radicand
The number under the root sign. In √8 the radicand is 8.

entire radical
A radical with no number in front, such as √45.

mixed radical
A number times a radical, such as 3√5.

simplest form (radical)
The radicand has no perfect-square factor other than 1.

like radicals
Radicals with the same radicand, such as 2√3 and 7√3. Only like radicals can be added or subtracted.

conjugates
A pair such as a + √b and a − √b. Their product a² − b has no radical.

Exact answers

A community garden in Thunder Bay wants a square herb bed with an area of exactly 8 m². How long is each side? The side is √8 m. A calculator shows 2.828427125…, and the decimals never end or repeat. Any decimal you write is a rounded approximation.

Mathematicians often prefer the exact answer. But √8 is not in its simplest form: it equals 2√2. In this unit, the zeros of many quadratic functions are radicals, like 1 + √5, so you need to simplify, add and multiply radicals with confidence. The sibling unit Functions and Equivalent Expressions also uses these skills for equivalent expressions; here we focus on what you need for quadratic functions.

The multiplication property

Try these on a calculator:

√4 × √9 = 2 × 3 = 6, and √36 = 6.
√2 × √8 = 1.41421… × 2.82842… = 4, and √16 = 4.

This always works for numbers that are not negative:

√(ab) = √a × √b, for a ≥ 0 and b ≥ 0.

Read from left to right, it lets you split a radical. Read from right to left, it lets you combine two radicals into one.

It does not work for addition: √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7.

Squares drawn on a dot grid. The small tilted square has area 2, so its side is √2. The large tilted square has area 4 × 4 − 4 × 2 = 8, and each of its sides is two small diagonals long, so √8 = 2√2.

Simplifying a radical

A radical is in simplest form when the number under the root sign (the radicand) has no perfect-square factor other than 1.

Find the largest perfect square that divides the radicand. Useful list: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
Split the radical: √(square × other) = √square × √other.
Take the square root of the perfect square.

A number written like 2√6 (a number times a radical) is a mixed radical. A number written like √24 is an entire radical.

Simplifying √72 in one pass: 36 is the largest perfect square that divides 72. If you use a smaller square first (√72 = √9 × √8 = 3√8), you must keep going: 3√8 = 3 × 2√2 = 6√2.

Worked example

Example 1: Simplify

Write in simplest form: (a) √24 (b) √72 (c) √200

(a) 24 = 4 × 6, and 6 has no square factor. √24 = √4 × √6 = 2√6.
(b) 72 = 36 × 2. √72 = √36 × √2 = 6√2.
(c) 200 = 100 × 2. √200 = √100 × √2 = 10√2.
Check (a) on a calculator: √24 ≈ 4.899 and 2√6 ≈ 2 × 2.449 = 4.899 ✓.

√24 = 2√6, √72 = 6√2, √200 = 10√2.

Worked example

Example 2: Mixed to entire, and comparing

Write 3√5 and 4√3 as entire radicals. Which is larger?

Move the number inside by squaring it: 3√5 = √9 × √5 = √45.
4√3 = √16 × √3 = √48.
√48 > √45, so 4√3 is larger (4√3 ≈ 6.93 and 3√5 ≈ 6.71).

3√5 = √45, 4√3 = √48, and 4√3 is larger.

Adding and subtracting radicals

You can only combine like radicals: radicals with the same radicand, just as 2x + 5x = 7x.

2√3 + 5√3 = 7√3.
√2 + √3 cannot be combined.

Often radicals only look unlike. Simplify first, then combine.

Worked example

Example 3: Simplify first, then add

Simplify (a) √12 + √27 (b) 5√2 − √18 + √50

(a) √12 = 2√3 and √27 = 3√3. So √12 + √27 = 2√3 + 3√3 = 5√3.
(b) √18 = 3√2 and √50 = 5√2. So 5√2 − 3√2 + 5√2 = 7√2.

(a) 5√3 (b) 7√2

Multiplying radicals

Multiply the numbers outside, multiply the radicands, then simplify: (a√b)(c√d) = ac√(bd).

For binomials, use the distributive property, just like (x + 2)(x − 3). Remember that √a × √a = a.

Worked example

Example 4: Products

Simplify (a) (2√3)(5√6) (b) (2 + √6)(3 − √12) (c) (1 + √5)(1 − √5)

(a) (2 × 5)√(3 × 6) = 10√18 = 10 × 3√2 = 30√2.
(b) Expand all four products: 2(3) + 2(−√12) + √6(3) + √6(−√12) = 6 − 2√12 + 3√6 − √72.
Simplify each radical: 2√12 = 2(2√3) = 4√3 and √72 = 6√2. The result is 6 − 4√3 + 3√6 − 6√2. No two terms are like, so this is the final answer.
(c) This is a difference of squares: 1² − (√5)² = 1 − 5 = −4. The radicals disappear! Pairs like 1 + √5 and 1 − √5 are called conjugates.

(a) 30√2 (b) 6 − 4√3 + 3√6 − 6√2 (c) −4

Worked example

Example 5: Dividing out a common factor

Simplify (6 + √12)/2. (Expressions like this come out of the quadratic formula.)

Simplify the radical first: √12 = 2√3, so the expression is (6 + 2√3)/2.
Both terms of the numerator have a factor of 2: (6 + 2√3)/2 = 2(3 + √3)/2.
Divide: 3 + √3.
Common mistake: dividing only the 6 by 2 to get 3 + 2√3. The fraction bar divides the whole numerator.

(6 + √12)/2 = 3 + √3

Hands-on activity

Measure a radical

You need: a ruler, a pencil and graph paper (or plain paper and a set square).

Draw a square with sides of 5 cm. Measure its diagonal as carefully as you can.
By the Pythagorean theorem the diagonal is √(5² + 5²) = √50 = 5√2 cm. Use a calculator to find 5√2. How close was your measurement?
On graph paper, draw the tilted square with corners at (2, 0), (4, 2), (2, 4) and (0, 2) (1 square = 1 cm). Find its area by taking the 4 by 4 square around it and subtracting the four corner triangles.
Measure one side. Compare with √8 = 2√2.

What to look for: the diagonal is about 7.1 cm (5√2 ≈ 7.07). The tilted square has area 16 − 4(2) = 8 cm², and its side measures about 2.8 cm (2√2 ≈ 2.83).

Just for fun

The decimal 2.8284271247461903… went on and on about how long it was. The radical just shrugged and said, "I can say all of that in three characters: 2√2."

Your turn

Simplify √20.

Simplify √75.

Simplify √98.

Write 2√7 as an entire radical.

Simplify 3√2 + √32.

Simplify (√3)(√15).

Expand and simplify (4 + √2)(3 − √2).

More practice: warm-up

Simplify √9 × √4.

Simplify √50.

Simplify √18.

Simplify 5√3 − 2√3.

Simplify √5 × √5.

More practice: on your own

Simplify √8 + √18 − √32.

Expand and simplify 2√3(√6 − 5).

Expand and simplify (2 + √6)(3 − √12).

Expand and simplify (5 − √3)(5 + √3).

Simplify (8 − √48)/4.

Challenge

Order from least to greatest without a calculator: 3√6, 4√3, 2√13, 5√2.

Show that x = 1 + √5 is a zero of f(x) = x² − 2x − 4.

A square has a diagonal of 10 cm. Find its exact side length and its area.

Check your understanding

Quiz

Which is √48 in simplest form?

Simplify √12 + √3.

For a ≥ 0 and b ≥ 0, which statement is always true?

Simplify (√6)(√10).

Write 3√5 as an entire radical.

Simplify (3 + √2)(3 − √2).

For parents and teachers

Big idea: Exact zeros of quadratic functions are often radicals, so this lesson builds radical skills (curriculum expectation A3.2: verify √ab = √a × √b and use it to simplify radicals and radical expressions, including sums, differences and products such as (2 + √6)(3 − √12)). The same expectation is also part of the sibling unit Functions and Equivalent Expressions.

How to help: Have your teen check every simplification with a calculator: √72 and 6√2 should give the same decimal. Quiz them on perfect squares to 144.

Common difficulties:

Not using the largest square factor, and stopping too early (3√8 is not simplest).
Adding unlike radicals (√2 + √3 ≠ √5).
Splitting a sum: √(a + b) is not √a + √b.
Dividing only one term of a numerator by the denominator.