Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt{\frac{225 x^{3}}{49 x}}\)

Step-by-Step Solution

Answer

\(\frac{15x}{7}\)

Step 1: Simplify the Expression Inside the Square Root

Begin by simplifying the expression inside the square root: \(\frac{225x^3}{49x}\). Divide both the numerator and the denominator by \(x\), which gives \(\frac{225x^2}{49}\).

Step 2: Simplify the Fraction Further

Notice that \(225\) and \(49\) are perfect squares. \(225 = 15^2\) and \(49 = 7^2\). So the fraction becomes \(\frac{15^2x^2}{7^2}\).

Step 3: Apply the Square Root

Apply the square root to both the numerator and the denominator: \(\sqrt{\frac{15^2x^2}{7^2}}\). This can be simplified by separating the square root into \(\frac{\sqrt{15^2x^2}}{\sqrt{7^2}}\).

Step 4: Simplify the Square Roots

The square root of \(15^2x^2\) is \(15x\), and the square root of \(7^2\) is \(7\). Thus, the expression simplifies to \(\frac{15x}{7}\).

Key Concepts

Square Roots

Square roots are fundamental elements in mathematics that involve finding a number which, when multiplied by itself, gives the original number. For instance, the square root of 36 is 6, because 6 times 6 equals 36. Square roots are often represented with the radical symbol \( \sqrt{} \). In an algebraic context, like \( \sqrt{16x^2} \), the square root is applied to both the number and the variable.

Key aspects to remember with square roots include:

• Only non-negative numbers have real square roots.
• The square root of a product \( ab \) is the product of the square roots \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \).
• You can often simplify expressions involving square roots by rewriting the numbers as a product of perfect squares.

By understanding square roots, you can simplify expressions more easily, as seen with \( \sqrt{\frac{225x^3}{49x}} \). Here, breaking down the expression inside the square root allows for further simplification.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both, contain algebraic expressions—these could be variables, coefficients, or both. Simplifying algebraic fractions involves reducing them similarly to numerical fractions.

To simplify an algebraic fraction:

• Identify common factors in the numerator and the denominator.
• Remove or cancel identical factors.
• Apply any additional simplification steps based on algebraic rules.

In our example, \( \frac{225x^3}{49x} \), the first step involved canceling out \( x \) from both the numerator and the denominator to simplify to \( \frac{225x^2}{49} \). This makes the expression manageable and ready for further simplification or application of other mathematical operations like square roots.

Perfect Squares

Perfect squares are numbers or expressions that result from squaring whole numbers. Recognizing perfect squares is critical in simplifying mathematical expressions, especially those involving square roots.

Perfect squares have the properties:

• They are always non-negative.
• Recognizable in the form \( n^2 \), where \( n \) is an integer or an algebraic term.
• Useful for simplifying square roots, as the square root of a perfect square yields a whole number or simplified expression.

In the exercise, \( 225 \) and \( 49 \) are perfect squares, as \( 225 = 15^2 \) and \( 49 = 7^2 \). This knowledge helps us break down the expression \( \sqrt{\frac{15^2x^2}{7^2}} \) and simplify it to a simpler form like \( \frac{15x}{7} \). Mastering the recognition of perfect squares is a strategic advantage in mathematics, particularly in working with radical expressions.