Problem 113 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{200 x^{7}}{2 x^{3}}} $$

Step-by-Step Solution

Answer

10x^{2}

Step 1 - Simplify the fraction inside the square root

The given expression is \ \ \ \( \sqrt{\frac{200 x^{7}}{2 x^{3}}}\) \ \ \ First, simplify the fraction inside the square root: \ \ \ \[ \frac{200 x^{7}}{2 x^{3}} = \frac{200}{2} \cdot \frac{x^{7}}{x^{3}} = 100 x^{4}\]

Step 2 - Apply the square root to the simplified expression

Next, apply the square root to the simplified expression: \ \ \ \[ \sqrt{100 x^{4}}\] \ \ \ Recall that \ \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) \, we can separate the terms as: \ \ \ \[ \sqrt{100} \cdot \sqrt{x^{4}}\]

Step 3 - Simplify the square root of each term

Simplify the square root of each term separately: \ \ \ \[ \sqrt{100} = 10 \] \ \ and \ \ \ \[ \sqrt{x^{4}} = x^{2}\] \ \ because \ \( x^{4} = (x^{2})^{2} \). So the square root of \ \( x^{4} \) is \ \( x^{2} \).

Step 4 - Combine the simplified terms

Combine the simplified terms back together: \ \ \ \[ 10 \cdot x^{2} = 10x^{2}\]

Key Concepts

simplifying radicals

Simplifying radicals involves breaking down a radical expression into its simplest form.
A radical often involves a square root, cube root, or higher roots.
To simplify a radical, follow these steps: identify and factor the number or expression under the radical, and then simplify each part separately.
Consider the example from the exercise: \( \sqrt{\frac{200 x^{7}}{2 x^{3}}}\)
First, simplify the fraction: \[ \frac{200 x^{7}}{2 x^{3}} = 100 x^{4} \]
Next, apply the square root to each part of the resulting expression: \[ \sqrt{100} \times \sqrt{x^{4}} = 10 \times x^2 = 10 x^2 \]
This method of breaking the expression into smaller parts makes it easier to manage.

fraction simplification

Simplifying fractions reduces them to their simplest form.
For this, divide both the numerator and the denominator by their greatest common divisor (GCD).
Look at the fraction in the exercise: \( \frac{200 x^{7}}{2 x^{3}} \)
Start by simplifying the coefficients: Divide 200 by 2, giving 100.
Next, simplify the variables by subtracting the exponents: \[ \frac{x^{7}}{x^{3}} = x^{7-3} = x^{4} \] Combine these results: \[ \frac{200 x^{7}}{2 x^{3}} = 100 x^{4} \]
This gives the fraction in its simplest form.

properties of square roots

Square roots have special properties that help simplify expressions.
Key properties include:

• \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)
• \( \sqrt{a^2} = a \) for non-negative a
• \( \sqrt{a/b} = \sqrt{a} / \sqrt{b} \) for b ≠ 0

Applying these to the exercise: \[ \sqrt{100 x^{4}} \rightarrow \sqrt{100} \times \sqrt{x^{4}} \]
Simplify each component:

• \sqrt{100} = 10 \
• \sqrt{x^{4}} = (x^{2}) = x^2 \

Combine these for the final solution: \[ 10 x^{2} \]

exponent rules

Understanding exponent rules is fundamental in algebra.
The key rules include:

• \( a^{m} \times a^{n} = a^{m+n} \)
• \( \(a^{m}\)^{n} = a^{m \times n} \)
• \( \frac{a^{m}}{a^{n}} = a^{m-n} \) for a ≠ 0
• \ (a^0 = 1) for all a ≠ 0

For the given exercise:
Simplifying the exponents in:
\[ \frac{ x^{7} }{ x^{3} } = x^{7-3} = x^{4} \]
Remembering these rules makes working with exponents much easier and is crucial for simplifying algebraic expressions correctly.