Mastering Square Roots of Decimals

Finding the square root of a decimal can feel intimidating at first glance. However, there is a simple “secret” to solving these: convert the decimal into a fraction.

By turning the decimal into a fraction with a denominator like 100, 10,000, or 1,000,000, you can take the square root of the numerator and denominator separately.

The Basic Formula

To solve these, we use the property:

Step-by-Step Examples

1. Small Decimals

When dealing with numbers less than 1, count the decimal places to determine your denominator.

• Example 1: Find $\sqrt{0.01}$
  $$\sqrt{0.01} = \sqrt{\frac{1}{100}} = \frac{\sqrt{1}}{\sqrt{100}} = \frac{1}{10} = 0.1$$
• Example 2: Find $\sqrt{0.0001}$
  $$\sqrt{0.0001} = \sqrt{\frac{1}{10000}} = \frac{\sqrt{1}}{\sqrt{10000}} = \frac{1}{100} = 0.01$$
• Example 3: Find $\sqrt{0.000016}$
  $$\sqrt{0.000016} = \sqrt{\frac{16}{1000000}} = \frac{4}{1000} = 0.004$$

2. Larger Decimals

The method works exactly the same way for numbers greater than 1.

• Example 4: Find $\sqrt{1.69}$
  $$\sqrt{1.69} = \sqrt{\frac{169}{100}} = \frac{\sqrt{169}}{\sqrt{100}} = \frac{13}{10} = 1.3$$
• Example 5: Find $\sqrt{33.64}$
  $$\sqrt{33.64} = \sqrt{\frac{3364}{100}} = \frac{\sqrt{3364}}{10} = \frac{58}{10} = 58$$

  Example 1: Finding $\sqrt{110.25}$

  When you encounter a larger decimal like $110.25$, the most efficient approach is to first eliminate the decimal point and then solve for the square root of the resulting large whole number.

  Step 1: Convert to a Fraction

  Since there are two decimal places, we can rewrite the expression as a fraction with $100$ in the denominator:

  $$\sqrt{110.25} = \sqrt{\frac{11025}{100}}$$

  Step 2: Simplify the Expression

  Apply the square root to both the numerator and the denominator separately:

  $$\frac{\sqrt{11025}}{\sqrt{100}} = \frac{\sqrt{11025}}{10}$$

  Step 3: Use Long Division for the Numerator

  To find the square root of $11,025$, we use the long division method:

  • Group the digits: Pair the digits from right to left: $\overline{1}\ \overline{10}\ \overline{25}$.

  • First Digit: $1 \times 1 = 1$. Subtract to get $0$ and bring down the next pair, $10$.

  • Second Digit: Double the quotient ($1 \times 2 = 2$). Since $20 \times 0$ is the largest product less than $10$, the next digit is $0$. Subtract $0$ and bring down $25$.

  • Third Digit: Double the current quotient ($10 \times 2 = 20$). We find that $205 \times 5 = 1025$.

  • Result: $\sqrt{11025} = 105$.

  Step 4: Final Decimal Calculation

  Now, plug the result back into your fraction:

  $$\frac{105}{10} = 10.5$$