Understanding the Square Root Problem

The question asks us to find the sum of three different square root values: $\sqrt{46.24}$, $\sqrt{0.4624}$, and $\sqrt{0.004624}$. We are given a helpful piece of information: $\sqrt{4624} = 68$. We can use this fact to calculate the values of the square roots involving decimals.

Calculating Square Roots with Decimals

When finding the square root of a number with a decimal, we can relate it back to the square root of the number without the decimal by considering the number of decimal places. The key is that the number of decimal places in the square root is half the number of decimal places in the original number under the root symbol.

Let's calculate each term separately:

Calculating $\sqrt{46.24}$

The number 46.24 has 2 decimal places. Since $\sqrt{4624} = 68$, the square root of 46.24 will have half the number of decimal places, which is $2/2 = 1$ decimal place.

So, $\sqrt{46.24} = 6.8$.

Alternatively, we can write this as:

$\sqrt{46.24} = \sqrt{\frac{4624}{100}} = \frac{\sqrt{4624}}{\sqrt{100}} = \frac{68}{10} = 6.8$

Calculating $\sqrt{0.4624}$

The number 0.4624 has 4 decimal places. The square root will have half the number of decimal places, which is $4/2 = 2$ decimal places.

Since $\sqrt{4624} = 68$, we need to place the decimal point two places from the right in 68.

So, $\sqrt{0.4624} = 0.68$.

Alternatively, using fractions:

$\sqrt{0.4624} = \sqrt{\frac{4624}{10000}} = \frac{\sqrt{4624}}{\sqrt{10000}} = \frac{68}{100} = 0.68$

Calculating $\sqrt{0.004624}$

The number 0.004624 has 6 decimal places. The square root will have half the number of decimal places, which is $6/2 = 3$ decimal places.

Since $\sqrt{4624} = 68$, we need to place the decimal point three places from the right in 68. We add a leading zero to achieve this.

So, $\sqrt{0.004624} = 0.068$.

Alternatively, using fractions:

$\sqrt{0.004624} = \sqrt{\frac{4624}{1000000}} = \frac{\sqrt{4624}}{\sqrt{1000000}} = \frac{68}{1000} = 0.068$

Summing the Calculated Square Roots

Now we need to find the sum of the three calculated square root values:

Sum = $\sqrt{46.24} + \sqrt{0.4624} + \sqrt{0.004624}$

Sum = $6.8 + 0.68 + 0.068$

Let's align the decimal points and add:

  6.800
  0.680
+ 0.068
-------
  7.548

The sum is 7.548.

Final Result and Option Matching

The calculated sum is 7.548. Let's compare this with the given options:

• Option 1: 7.548
• Option 2: 7.854
• Option 3: 7.458
• Option 4: 7.648

Our calculated value, 7.548, matches Option 1.

Additional Information: Properties of Square Roots

Understanding the properties of square roots can simplify many calculations involving roots. Some important properties include:

• Product Property: The square root of a product is the product of the square roots. Mathematically, for non-negative numbers a and b, $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$.
• Quotient Property: The square root of a quotient is the quotient of the square roots. Mathematically, for a non-negative number a and a positive number b, $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. This property was used in our calculations above by converting decimal numbers to fractions. For example, $\sqrt{46.24} = \sqrt{\frac{4624}{100}} = \frac{\sqrt{4624}}{\sqrt{100}}$.
• Squaring a Square Root: Squaring a non-negative number's square root gives the original number. $(\sqrt{a})^2 = a$ for $a \ge 0$.

These properties are fundamental when working with square roots and other radicals.