Approximating Square Roots Study Guide

Fundamentals of Perfect Squares and Square Roots

• Radical expressions represent principal square roots, where the symbol $\sqrt{}$ indicates finding the non-negative factor that, when multiplied by itself, equals the radicand.
• Exact values for perfect square radicals:
  • $\sqrt{0} = 0$
  • $\sqrt{1} = 1$
  • $\sqrt{4} = 2$
  • $\sqrt{9} = 3$
  • $\sqrt{16} = 4$
  • $\sqrt{25} = 5$
  • $\sqrt{36} = 6$
  • $\sqrt{49} = 7$
  • $\sqrt{64} = 8$
  • $\sqrt{81} = 9$
  • $\sqrt{100} = 10$
  • $\sqrt{121} = 11$
  • $\sqrt{144} = 12$
• Non-perfect square radicands result in irrational numbers, which cannot be expressed as simple integer ratios and must be approximated.
• Ordering integers and radicals along a number line:
  • Negative integers: $-14$, $-11$, $-5$, $-4$, $-3$, $-2$, $-1$
  • Position $0$ corresponds to $\sqrt{0} = 0$
  • Position $1$ corresponds to $\sqrt{1} = 1$
  • Position $2$ corresponds to $\sqrt{4} = 2$
  • Position $3$ corresponds to $\sqrt{9} = 3$
  • Radical $\sqrt{14}$ is situated between $\sqrt{9}$ ($3$) and $\sqrt{25}$ ($5$)
  • Position $5$ corresponds to $\sqrt{25} = 5$
  • Radical $\sqrt{27}$ is situated between $\sqrt{25}$ ($5$) and $\sqrt{36}$ ($6$)
  • Radical $\sqrt{34}$ is situated between $\sqrt{25}$ ($5$) and $\sqrt{36}$ ($6$)
  • Position $7$ corresponds to $\sqrt{49} = 7$
  • Radical $\sqrt{57}$ is situated between $\sqrt{49}$ ($7$) and $\sqrt{64}$ ($8$)
  • Position $8$ corresponds to $\sqrt{64} = 8$
  • Position $9$ corresponds to $\sqrt{81} = 9$
  • Position $10$ corresponds to $\sqrt{100} = 10$
  • Radical $\sqrt{135}$ is situated between $\sqrt{121}$ ($11$) and $\sqrt{144}$ ($12$)
  • Position $12$ corresponds to $\sqrt{144} = 12$

Procedure for Approximating Irrational Square Roots

• Step 1: Identify Bounding Consecutive Whole Numbers

  • Determine the lower perfect square and the bigger (upper) perfect square that enclose the given irrational radicand.
  • Take the principal square root of both perfect squares to establish the two consecutive whole numbers bounding the target value.
  • Identify the lower whole number as the integer component of the approximation.

• Step 2: Construct the Linear Interpolation Ratio

  • Formulate the fractional ratio comparing the distance from the lower perfect square to the total interval between the two bounding perfect squares.
  • Formula for the fractional component:     $\text{Fractional Part} = \frac{\text{Irrational Radicand} - \text{Lower Perfect Square}}{\text{Bigger Perfect Square} - \text{Lower Perfect Square}}$

• Step 3: Convert to Mixed Number and Decimal Form

  • Combine the lower bounding whole number with the ratio from Step 2 to form a mixed number:     $\text{Mixed Number} = \text{Lower Whole Number} + \frac{\text{Irrational Radicand} - \text{Lower Perfect Square}}{\text{Bigger Perfect Square} - \text{Lower Perfect Square}}$
  • Perform long division on the fractional portion to compute its decimal expansion.
  • Add the decimal fraction to the integer component and round the result to the nearest tenth or hundredth as requested.

Worked Examples of Square Root Approximations

• Example 1: Approximation of $\sqrt{27}$

  • Step 1: Find bounding perfect squares.
  • Bounding lower perfect square: $\sqrt{25} = 5$
  • Bounding upper perfect square: $\sqrt{36} = 6$
  • Result: $\sqrt{27}$ lies between whole numbers $5$ and $6$
  • Step 2: Calculate ratio ratio.
  • Numerator: $27 - 25 = 2$
  • Denominator: $36 - 25 = 11$
  • Fraction: $\frac{2}{11}$
  • Step 3: Convert to mixed number and decimal.
  • Mixed number expression: $5\frac{2}{11}$
  • Long division computation for $\frac{2}{11}$:
    • $2.00 \div 11$
    • $20 - 11 = 9$
    • $90 - 88 = 2$
    • Quotient: $0.1818... \approx 0.18$
  • Unrounded decimal expression: $5.18$
  • Final value rounded to the nearest tenth: $5.2$ (or $5.10$ / $5.18$ in intermediate steps)

• Example 2: Approximation of $\sqrt{57}$

  • Step 1: Find bounding perfect squares.
  • Bounding lower perfect square: $\sqrt{49} = 7$
  • Bounding upper perfect square: $\sqrt{64} = 8$
  • Result: $\sqrt{57}$ lies between whole numbers $7$ and $8$
  • Step 2: Calculate ratio.
  • Numerator: $57 - 49 = 8$
  • Denominator: $64 - 49 = 15$
  • Fraction: $\frac{8}{15}$
  • Step 3: Convert to mixed number and decimal.
  • Mixed number expression: $7\frac{8}{15}$
  • Long division computation for $\frac{8}{15}$:
    • $8.00 \div 15$
    • $80 - 75 = 5$
    • $50 - 45 = 5$
    • Quotient: $0.533... \approx 0.53$
  • Decimal expression: $7.53$

• Example 3: Approximation of $\sqrt{135}$

  • Step 1: Find bounding perfect squares.
  • Bounding lower perfect square: $\sqrt{121} = 11$
  • Bounding upper perfect square: $\sqrt{144} = 12$
  • Result: $\sqrt{135}$ lies between whole numbers $11$ and $12$
  • Step 2: Calculate ratio.
  • Numerator: $135 - 121 = 14$
  • Denominator: $144 - 121 = 23$
  • Fraction: $\frac{14}{23}$
  • Step 3: Convert to mixed number and decimal.
  • Mixed number expression: $11\frac{14}{23}$
  • Decimal expression rounded to nearest tenth: $11.6$