Mathematics 7: Perfect Roots, Square Roots, Cube Roots, and Approximations

Course Overview and Lesson Identification

• Subject: Mathematics 7
• Unit and Lessons Covered:
  • Unit 5, Lesson 1: Perfect Roots
  • Unit 5, Lesson 2: Square Root and Cube Root of a Number
• Instructional Designer / Educator: Ethel Lorraine Justiniane, LPT

Perfect Squares

• Definition: A perfect square is a number that can be written as the product of two equal numbers.
• Fundamental Multiplication Examples:
  • $2 \times 2 = 4$
  • $3 \times 3 = 9$
  • $5 \times 5 = 25$

Perfect Cubes

• Definition: A perfect cube is a number that can be written as the product of three equal numbers.
• Multiplication Examples and Decomposition:
  • $2 \times 2 \times 2 = 8$
  • $3 \times 3 \times 3 = 27$
    • First stage: $3 \times 3 = 9$
    • Second stage: $9 \times 3 = 27$
  • $5 \times 5 \times 5 = 125$
    • First stage: $5 \times 5 = 25$
    • Second stage: $25 \times 5 = 125$

Table of Perfect Squares and Perfect Cubes (Numbers 1 to 20)

• Integers 1 through 10:
  • Number $1$: Square = $1$, Cube = $1$
  • Number $2$: Square = $4$, Cube = $8$
  • Number $3$: Square = $9$, Cube = $27$
  • Number $4$: Square = $16$, Cube = $64$
  • Number $5$: Square = $25$, Cube = $125$
  • Number $6$: Square = $36$, Cube = $216$
  • Number $7$: Square = $49$, Cube = $343$
  • Number $8$: Square = $64$, Cube = $512$
  • Number $9$: Square = $81$, Cube = $729$
  • Number $10$: Square = $100$, Cube = $1{,}000$
• Integers 11 through 20:
  • Number $11$: Square = $121$, Cube = $1{,}331$
  • Number $12$: Square = $144$, Cube = $1{,}728$
  • Number $13$: Square = $169$, Cube = $2{,}197$
  • Number $14$: Square = $196$, Cube = $2{,}744$
  • Number $15$: Square = $225$, Cube = $3{,}375$
  • Number $16$: Square = $256$, Cube = $4{,}096$
  • Number $17$: Square = $289$, Cube = $4{,}913$
  • Number $18$: Square = $324$, Cube = $5{,}832$
  • Number $19$: Square = $361$, Cube = $6{,}859$
  • Number $20$: Square = $400$, Cube = $8{,}000$

Square Roots and Cube Roots Definitions

• Square Root of a Number:
  • A number that, when used as a factor two times, gives the original number.
  • The radical symbol for square root is $\sqrt{\phantom{x}}$.
• Cube Root of a Number:
  • A number that, when used as a factor three times, gives the original number.
  • The symbol for cube root is $\sqrt[3]{\phantom{x}}$.
  • The number $3$ in $\sqrt[3]{\phantom{x}}$ represents the index of the root.

Evaluating Exact Roots

• Example 1: Evaluating $\sqrt{144}$
  • Identity: $12 \times 12 = 144$
  • Solution: $\sqrt{144} = 12$
• Example 2: Evaluating $\sqrt{49}$
  • Identity: $7 \times 7 = 49$
  • Solution: $\sqrt{49} = 7$
• Example 3: Evaluating $\sqrt[3]{729}$
  • Identity: $9 \times 9 \times 9 = 81 \times 9 = 729$
  • Solution: $\sqrt[3]{729} = 9$
• Example 4: Evaluating $\sqrt[3]{125}$
  • Identity: $5 \times 5 \times 5 = 25 \times 5 = 125$
  • Solution: $\sqrt[3]{125} = 5$

Rational and Irrational Numbers

• Rational Numbers:
  • Numbers that can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.
• Irrational Numbers:
  • Numbers that cannot be expressed as a simple fraction $\frac{a}{b}$.
  • Their decimal representations are non-terminating and non-repeating.
  • Classic radical example: $\sqrt{2}$

Approximating Square Roots Using Heron's Method

• Purpose: To approximate square roots of non-perfect squares.
• Sequential Procedural Steps:
  1. Take a first guess.
  2. Divide the original number by the guess.
  3. Add the guess to the quotient.
  4. Divide the sum by $2$.
  5. Use the value as the new guess and repeat the process.
• Comprehensive Worked Example: Approximate $\sqrt{10}$
  • Step 1: Determine the First Guess
    • Identify the closest surrounding perfect squares to $10$:
    • $\sqrt{9} < \sqrt{10} < \sqrt{16}$
    • $3 < \sqrt{10} < 4$
    • Since $10$ is closer to $9$ than to $16$, choose initial guess $= 3$.
  • Step 2: Divide the Original Number by the Guess
    • Calculate $10 \div 3$
    • Long division process: $10.00 \div 3$
    • $10 - 9 = 1$
    • Bring down $0$ to make $10$, $10 - 9 = 1$
    • Bring down $0$ to make $10$, $10 - 9 = 1$
    • Quotient $= 3.33$
  • Step 3: Add the Guess to the Quotient
    • Calculate $3 + 3.33 = 6.33$
  • Step 4: Divide the Sum by 2
    • Calculate $6.33 \div 2$
    • Long division setup: $2 \overline{)6.330}$
    • $6 - 6 = 0$
    • Bring down $3$: $3 - 2 = 1$
    • Bring down $3$: $13 - 12 = 1$
    • Bring down $0$: $10 - 10 = 0$
    • Resulting value $= 3.165$
  • Final Approximate Value:
    • $\sqrt{10} \approx 3.165$

Approximating Cube Roots Using Linear Approximation

• Purpose: To approximate cube roots of non-perfect cubes.
• Sequential Procedural Steps:
  1. Determine the two perfect cubes closest to the given number.
  2. Subtract the smaller perfect cube from the original number (yields the numerator).
  3. Subtract the two perfect cubes (yields the denominator).
  4. Divide the value to the cube root of the smaller perfect cube. Add that to the cube root of the smaller number.
• Comprehensive Worked Example: Approximate $\sqrt[3]{10}$
  • Step 1: Determine Closest Perfect Cubes
    • Identify bounding perfect cubes:
    • $\sqrt[3]{8} < \sqrt[3]{10} < \sqrt[3]{27}$
    • $2 < \sqrt[3]{10} < 3$
    • The smaller perfect cube is $8$, with cube root $= 2$.
  • Step 2: Compute Numerator
    • Subtract the smaller perfect cube from the original number:
    • $10 - 8 = 2$
    • Numerator $= 2$
  • Step 3: Compute Denominator
    • Subtract the two perfect cubes:
    • $27 - 8 = 19$
    • Denominator $= 19$
  • Step 4: Perform Division and Decimal Conversion
    • Form the fractional adjustment: $\frac{2}{19}$
    • Long division: $19 \overline{)2.0}$
    • $20 - 19 = 1$
    • Decimal value $\approx 0.1$
  • Step 5: Final Addition
    • Add the quotient to the cube root of the smaller perfect cube:
    • $0.1 + 2 = 2.1$
  • Final Approximate Value:
    • $\sqrt[3]{10} \approx 2.1$