Simplify Square Root of 147 Calculator

Simplifying square roots is a fundamental math skill that helps in algebra, calculus, and many other areas of mathematics. This guide explains how to simplify √147, including the step-by-step process and common pitfalls to avoid.

How to Simplify √147

Simplifying a square root involves expressing it in terms of a product of square roots of perfect squares and other factors. For √147, we'll follow these steps:

Factor the number under the square root into perfect squares and other factors.
Separate the square root of the perfect square from the remaining factors.
Simplify the expression.

Key Point: A perfect square is an integer that is the square of another integer (e.g., 1, 4, 9, 16, 25, etc.).

Step-by-Step Simplification

Let's simplify √147 step by step:

Factor 147: First, find the factors of 147. The prime factorization of 147 is 3 × 7 × 7 or 3 × 7².
Identify perfect squares: From the factors, 7² is a perfect square (7 × 7 = 49).
Separate the square root: Using the property √(a × b) = √a × √b, we can write √147 as √(49 × 3).
Simplify: √49 is 7, so √147 = 7 × √3.

√147 = √(49 × 3) = √49 × √3 = 7√3

Therefore, the simplified form of √147 is 7√3.

Examples

Let's look at a few more examples to reinforce the concept:

√72: Factor into 36 × 2. √36 is 6, so √72 = 6√2.
√192: Factor into 64 × 3. √64 is 8, so √192 = 8√3.
√200: Factor into 100 × 2. √100 is 10, so √200 = 10√2.

Tip: Always look for the largest perfect square factor to simplify the square root most effectively.

FAQ

What is the simplified form of √147?: The simplified form of √147 is 7√3.
How do I simplify a square root?: To simplify a square root, factor the number under the root into perfect squares and other factors, then separate the square root of the perfect square from the remaining factors.
Can all square roots be simplified?: Not all square roots can be simplified. If the number under the square root is a prime number or doesn't have any perfect square factors other than 1, the square root is already in its simplest form.
What is the difference between simplifying and rationalizing a square root?: Simplifying a square root involves expressing it in terms of a product of a perfect square and another square root. Rationalizing involves eliminating the square root from the denominator of a fraction.
How can I check if my simplification is correct?: To verify your simplification, square the simplified form and check if it equals the original number under the square root. For example, (7√3)² = 49 × 3 = 147, which matches the original number.