Problem 46 - College Algebra

Question

Simplify each expression. $$\sqrt{490 b c^{2}}$$

Step-by-Step Solution

Answer

\(7c\sqrt{10b}\)

Step 1: Identify Perfect Squares

Identify any perfect square factors inside the square root. The number 490 can be factored as \(490 = 49 imes 10\). Here, 49 is a perfect square. For the variables, \(c^2\) is also a perfect square.

Step 2: Simplifying the Expression

Rewrite the expression by taking the square root of the perfect squares separately: \(\sqrt{490bc^2} = \sqrt{49 \times 10 \times b \times c^2}\). Take the square roots: \(\sqrt{49} = 7\) and \(\sqrt{c^2} = c\). Multiply the results inside the square root: \(7c\sqrt{10b}\).

Step 3: Final Expression

The simplified form of the expression is \(7c\sqrt{10b}\) where all possible square roots have been taken and the expression is in its simplest form.

Key Concepts

Perfect Squares

A perfect square is a number that results from multiplying an integer by itself. Recognizing perfect squares can greatly simplify calculations, especially when dealing with square root expressions.
For example:

• The number 49 is a perfect square because it is equal to \(7 \times 7\).
• Similarly, in terms of algebra, \(c^2\) is a perfect square because it is \(c \times c\).

Understanding perfect squares is crucial because when simplifying square roots, you can take the square root of these numbers or expressions directly. This helps in reducing complex expressions into simpler forms.
For instance, in the original problem \(\sqrt{490b c^2}\), recognizing 49 as a perfect square allows you to simplify it to 7 outside the square root. Likewise, \(c^2\) simplifies to \(c\). Being able to quickly identify these structures is key in manipulating algebraic expressions efficiently.

Algebraic Expressions

An algebraic expression combines numbers and variables using operations such as addition, subtraction, multiplication, and division. They can include constants (like numbers) and variables (like \(b\) or \(c\)).
To illustrate:

• The expression inside the square root \(\sqrt{490 b c^2}\) is an algebraic expression.
• It combines the number 490 with variables \(b\) and \(c^2\).

Simplifying an algebraic expression can involve grouping like terms and factoring out common variables or numbers. In our problem, we look at the expression as a product of simpler parts: identifying perfect squares and non-perfect square variables.
Breaking down complex expressions into simpler, more manageable parts can make it much easier to work with them, especially when you're required to simplify or solve them in mathematics.

Variables in Square Roots

Variables within square roots can be tricky, but with some understanding of basic properties of powers and roots, they become easier to manage. When a variable is squared, as with \(c^2\), and placed under a square root, it simplifies neatly to the variable itself: \(\sqrt{c^2} = c\). This is due to the rule that \(\sqrt{x^2} = x\).
Here are some key points:

• If a variable's exponent is a perfect square, its square root will simplify easily.
• For example, \(\sqrt{b^2} = b\) and \(\sqrt{c^4} = c^2\).

If variables do not have exponents that are perfect squares, like \(b\) in our expression, they remain inside the square root.
Therefore, in \(\sqrt{10b}\), \(b\) stays under the root. It's essential to distinguish which parts of the expression can be simplified based on whether the exponents allow for perfect squares. Understanding these properties allows for broader applications across various math problems where simplification is necessary.