91Ó°ÊÓ

Square roots are mathematical operations that find a number which, when multiplied by itself, results in the original number under the square root sign. They are represented by the radical symbol \( \sqrt{} \). Breaking numbers inside square roots into prime factors simplifies calculations. In our exercise, we use this simplification method to make the expression more manageable.

For example:

• The square root of 490 is broken down using prime factorization into its components: \(2*5*7^2\). Taking out perfect squares, this becomes \(7\sqrt{10}\).
• Similarly, for \(\sqrt{360y^3}\), prime factorization gives \(2^4*3^2*5*y^3\). This simplifies to \(12\sqrt{5y^3}\) after factoring out and simplifying the perfect squares.

Understanding how to manipulate square roots by using prime factorization is a powerful tool for simplifying algebraic expressions.

Prime factorization involves breaking down a composite number into the product of its prime numbers. This is a key step for simplifying square roots in algebraic expressions. Prime numbers are the building blocks of all natural numbers and include 2, 3, 5, 7, 11, etc., where each of these numbers has only two factors: one and itself.

Consider how this works in simplifying our expression:

• For 490, the prime factorization is \(2*5*7^2\). Recognizing and grouping this into perfect squares aids in simplifying square roots.
• In the case of 360, using \(2^4*3^2*5\) makes it feasible to extract \(2^2\) and \(3\) as outside constant multipliers, thereby simplifying the terms within the roots.

Efficiently breaking down numbers into prime factors helps streamline solving problems involving square roots and other more complex algebraic manipulations.

Algebraic expressions are combinations of variables and constants connected by arithmetic operations like addition, multiplication, and subtraction. Simplifying these expressions involves reducing them to a form that is easier to work with, which often includes consolidating like terms and simplifying mathematical operations where possible.

In our given expression, \(y \sqrt{490y} - 2\sqrt{360y^3}\), simplification begins by cleaning up the square roots through factorization and then addressing the coefficients:

• First, factor out square roots by identifying and extracting perfect squares from the prime factors.
• Then, multiply the constants and variables outside the square roots accordingly: this leads to simplified terms like \(7y\sqrt{10y}\) and \(12\sqrt{5y^3}\).
• Finally, check if any like terms can be combined to further simplify the expression.

Recognizing and correctly managing algebraic expressions mean using fundamental operations alongside factoring and simplifying radical expressions, making problems easier to solve or evaluate.