الرياضيات (بالإنجليزية) الصف الثاني الإعدادي، الأسبوع 3: Adding, subtracting and multiplying radical terms; properties of real-number operations

Radicals

الأسبوع 3 · العمليات على الجذور: الحدود الجذرية المتشابهة والمعكوسات والمرافق وقوانين الجذور التربيعية والتكعيبية · 3 دروس

Adding, subtracting and multiplying radical terms; properties of real-number operations

Mathematics Prep 2 Term 1, Unit 1, Lessons 3–4

GoalAdd, subtract and multiply radical terms, and use the properties of real numbers to shorten the work.

New words

radical term · a term such as 4√3: a number times a rootlike radicals · radical terms with the same number under the root, such as 4√3 and √3radicand · the number under the root signdistributive property · a(b + c) = ab + ac

Explanation

Radical terms behave like algebra terms. 4√3 and √3 are like radicals (same radicand 3), so you add the coefficients: 4√3 + √3 = 5√3 and −4√3 + √3 = −3√3. √2 and √5 are not like, so √2 + √5 cannot be combined, and √2 + √3 is certainly not √5. To multiply, remember (√a)² = a, so √3 × √3 = 3 and −4√3 × √3 = −4 × 3 = −12. The usual properties of real numbers still hold: addition and multiplication are commutative and associative, 0 and 1 are the identities, and multiplication distributes over addition. They let you open brackets with roots: (√7 − 1)(2√7 + 3) = 14 + 3√7 − 2√7 − 3 = 11 + √7.

Rule	Example
add like radicals	4√3 + √3 = 5√3
unlike radicals stay apart	√2 + √5
a root times itself	(√3)² = 3
open brackets	(√5 + 1)(√5 − 1) = 4

Worked examples

Simplify 6√2 − 2√2 + 3√5 + √5. Collect like radicals: (6 − 2)√2 + (3 + 1)√5 = 4√2 + 4√5. The two terms are unlike, so this is the final answer.

ƒFormula

2√3(4 + √3) = 8√3 + 6

2√3 × √3 = 2 × 3 = 6

Expand 2√3(4 + √3) − 4√3. Distribute: 8√3 + 2√3 × √3 − 4√3 = 8√3 + 6 − 4√3 = 6 + 4√3.

A rectangle has length 2√3 + 1 and width √3 (in cm). Its area is √3(2√3 + 1) = 6 + √3 cm². Its perimeter is 2(2√3 + 1 + √3) = 6√3 + 2 cm.

Class activity

Check with a calculator. Is √2 + √3 equal to √5? Find the decimals of both sides. Then check that 4√3 + √3 and 5√3 have the same decimal value. What do you notice?

Practice

Simplify 5√2 − 2√2 + √2.

Can 3√5 + 2√3 be simplified? Explain.

Find 3√6 × 2√6.

Expand (√5 + 2)(√5 − 2).

Expand 2√3(√3 + 5).