Radical Products Notes
Radical Products
Guided Structural Notes // Unit 4: Radicals
Name:
Date:
Phase 1: Foundation Recall
Perfect Squares (1-12)
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
Perfect Cubes (1-5)
1, 8, 27, 64, 125
Phase 2: The Product Property
To multiply two radicals with the same index, multiply the numbers inside.
\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \]
\[ \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b} \]
Phase 3: The Assembly Process
1
Outside First
Multiply coefficients (numbers on the outside).
2
Inside Second
Multiply radicands (numbers inside roots).
3
Simplify
Simplify using division to find perfect power factors.
Phase 4: Field Testing (Examples)
Example 1: Square Roots
\[ \sqrt{8} \cdot \sqrt{10} = \]
MULTIPLY: $\sqrt{\phantom{xxx}}$ DIVIDE: $\phantom{xxx} \div \phantom{x} = $
Example 2: Cube Roots
\[ \sqrt[3]{12} \cdot \sqrt[3]{6} = \]
MULTIPLY: $\sqrt[3]{\phantom{xxx}}$ DIVIDE: $\phantom{xxx} \div \phantom{x} = $
Example 3: Coefficients
\[ 3\sqrt{6} \cdot 4\sqrt{12} = \]
OUTSIDE: $\phantom{xx} \cdot \phantom{xx} = $ INSIDE: $\sqrt{\phantom{xxx}} = \dots$
Example 4: Large Cube Root
\[ \sqrt[3]{10} \cdot \sqrt[3]{50} = \]
MULTIPLY: $\sqrt[3]{\phantom{xxx}}$ DIVIDE: $\phantom{xxx} \div \phantom{x} = $
Phase 5: Self-Check Diagnostics
01
\[ \sqrt{20} \cdot \sqrt{5} \]
02
\[ 2\sqrt{14} \cdot \sqrt{2} \]
03
\[ \sqrt[3]{16} \cdot \sqrt[3]{4} \]
Radical Products Answer Key
Field Operations
Answer Key // Radical Products Worksheet
STATUS: VERIFIED
01
\( \sqrt{10} \cdot \sqrt{15} \)
\( 5\sqrt{6} \)
02
\( \sqrt{12} \cdot \sqrt{14} \)
\( 2\sqrt{42} \)
03
\( 2\sqrt{18} \cdot \sqrt{6} \)
\( 12\sqrt{3} \)
04
\( \sqrt{40} \cdot \sqrt{20} \)
\( 20\sqrt{2} \)
05
\( \sqrt[3]{4} \cdot \sqrt[3]{10} \)
\( 2\sqrt[3]{5} \)
06
\( \sqrt[3]{12} \cdot \sqrt[3]{9} \)
\( 3\sqrt[3]{4} \)
07
\( 2\sqrt[3]{25} \cdot \sqrt[3]{10} \)
\( 10\sqrt[3]{2} \)
08
\( \sqrt[3]{54} \cdot \sqrt[3]{2} \)
\( 3\sqrt[3]{4} \)
09
\( \sqrt[3]{16} \cdot \sqrt[3]{4} \)
\( 4 \)
10
\( 3\sqrt[3]{24} \cdot \sqrt[3]{3} \)
\( 6\sqrt[3]{9} \)
11
\( \sqrt[3]{36} \cdot \sqrt[3]{6} \)
\( 6 \)
12
\( \sqrt[3]{45} \cdot \sqrt[3]{3} \)
\( 3\sqrt[3]{5} \)
13
\( \sqrt{24} \cdot \sqrt{21} \)
\( 6\sqrt{14} \)
14
\( \sqrt[3]{20} \cdot \sqrt[3]{25} \)
\( 5\sqrt[3]{4} \)
15
\( 4\sqrt{12} \cdot 5\sqrt{6} \)
\( 120\sqrt{2} \)
Radical Products Practice Worksheet
Field Operations
Multiplication Practice // Sector 01: Square Roots
OPERATOR:
DATE:
Operational Directive
Multiply the radical expressions. Use long division to extract perfect factors. Simplify completely.
01
\[ \sqrt{10} \cdot \sqrt{15} \]
02
\[ \sqrt{12} \cdot \sqrt{14} \]
03
\[ 2\sqrt{18} \cdot \sqrt{6} \]
04
\[ \sqrt{40} \cdot \sqrt{20} \]
Sector 02: Cube Root Operations
Operational Log P. 02
05
\[ \sqrt[3]{4} \cdot \sqrt[3]{10} \]
06
\[ \sqrt[3]{12} \cdot \sqrt[3]{9} \]
07
\[ 2\sqrt[3]{25} \cdot \sqrt[3]{10} \]
08
\[ \sqrt[3]{54} \cdot \sqrt[3]{2} \]
Sector 03: Advanced Cube Roots
Operational Log P. 03
09
\[ \sqrt[3]{16} \cdot \sqrt[3]{4} \]
10
\[ 3\sqrt[3]{24} \cdot \sqrt[3]{3} \]
11
\[ \sqrt[3]{36} \cdot \sqrt[3]{6} \]
12
\[ \sqrt[3]{45} \cdot \sqrt[3]{3} \]
Sector 04: Mixed Review
Operational Log P. 04
13
\[ \sqrt{24} \cdot \sqrt{21} \]
14
\[ \sqrt[3]{20} \cdot \sqrt[3]{25} \]
15
\[ 4\sqrt{12} \cdot 5\sqrt{6} \]
Mission Complete
Simplify all results completely.
CODE: RAD-MULT-4PAGE // FINAL
