Square Root of 49/64

The square root is the inverse of the square of the number. 49/64 is a perfect square fraction. The square root of 49/64 is expressed in both radical and exponential form. In radical form, it is expressed as √(49/64), whereas (49/64)^(1/2) in exponential form. √(49/64) = 7/8, which is a rational number because it can be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method works well for perfect square numbers, including fractions. Let's discuss the methods commonly used to find square roots:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 49/64 is broken down into its prime factors.

Step 1: Finding the prime factors of 49 and 64 49 can be factored as 7 x 7 64 can be factored as 2 x 2 x 2 x 2 x 2 x 2

Step 2: Since 49 and 64 are both perfect squares, the square root can be directly obtained by taking the square root of the numerator and the denominator separately. √(49/64) = √49 / √64 = 7/8

The long division method is useful for non-perfect square numbers but can also verify perfect squares. Here is how to find the square root using the long division method, step by step:

Step 1: Write the number 49/64 in decimal form, which is 0.765625.

Step 2: Start by grouping digits in pairs from right to left in the decimal.

Step 3: Find the largest integer whose square is less than or equal to the first group. Here, the first group is 0.76, and 0.8 works because 0.8 x 0.8 = 0.64.

Step 4: Subtract, bring down the next group, and repeat the process for each pair.

Step 5: Continue the process until you reach an accurate value. For 0.765625, the square root is exactly 0.875, which confirms our previous result.

Approximation method is another method for finding square roots and is useful for non-perfect squares. However, for fractions like 49/64, we can directly find an exact result:

Step 1: Identify closest perfect squares for the numerator and the denominator.

Step 2: Since 49 and 64 are perfect squares, this method confirms that √49 = 7 and √64 = 8. Thus, √(49/64) = 7/8, which is an exact result.

• Square root: A square root is the inverse of a square. Example: 4² = 16 and the inverse of the square is the square root, that is √16 = 4.

• Rational number: A rational number is a number that can be written in the form of p/q, where q is not equal to zero, and p and q are integers.

• Principal square root: A number has both positive and negative square roots; however, it is always a positive square root that is more prominent due to its uses in the real world. That is why it is also known as a principal square root.

• Fraction: A fraction represents a part of a whole, expressed as a numerator divided by a denominator. For example, 7/8 is a fraction.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 49 and 64 are perfect squares because they are 7² and 8², respectively.