Problem 140 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{50 r^{5} s^{2}}{128 r^{2} s^{5}}} $$

Step-by-Step Solution

Answer

\[ \frac{5 \times r^{1.5}}{8 \times s^{1.5}} \]

Step 1: Simplify the Radicand

Start by simplifying the expression under the square root. Combine like terms. The expression becomes: \[ \frac{50 r^{5} s^{2}}{128 r^{2} s^{5}} \]

Step 2: Simplify Constants

Simplify the numerical part of the fraction. Divide 50 by 128 to simplify the fraction: \[ \frac{50}{128} = \frac{25}{64} \]

Step 3: Simplify Variables

Next, simplify the variables by subtracting the exponents in the numerator and the denominator: \[ r^{5} / r^{2} = r^{5-2} = r^{3} \] and \[ s^{2} / s^{5} = s^{2-5} = s^{-3} = \frac{1}{s^{3}} \]

Step 4: Combine Results

Combine the simplified constants and variables: \[ \frac{25 r^{3}}{64 s^{3}} \]

Step 5: Simplify the Square Root

Now, take the square root of the simplified expression: \[ \frac{25 r^{3}}{64 s^{3}} = \frac{\frac{5}{8} \times r^{3/2}}{s^{3/2}} \]

Step 6: Express Final Answer

Simplify the final expression: \[ \frac{5 r^{3/2}}{8 s^{3/2}} = \frac{5 \times r^{1.5}}{8 \times s^{1.5}} \]

Key Concepts

Fraction Simplification

When simplifying fractions, it's essential to reduce them to their simplest form. This involves dividing both the numerator (top number) and the denominator (bottom number) by their greatest common factor (GCF). In our example, the fraction under the square root is \(\frac{50 r^5 s^2}{128 r^2 s^5} \). Start by focusing on the numerical coefficients and simplify \(\frac{50}{128}\). We find that both 50 and 128 can be reduced by dividing by 2 until they can't be divided further:

• 50 ÷ 2 = 25
• 128 ÷ 2 = 64

Thus, \(\frac{50}{128}\) simplifies to \(\frac{25}{64}\).

Variable Exponents

Variable exponents follow the rules of exponents for division. When you divide like bases, you subtract the exponent in the denominator from the exponent in the numerator. For the variables in our exercise, you handle them separately:

• For \(\frac{r^5}{r^2}\), you subtract the exponents: \(\frac{r^5}{r^2} = r^{5-2} = r^3\).
• For \(\frac{s^2}{s^5}\), subtract the exponents: \(\frac{s^2}{s^5} = s^{2-5} = s^{-3}\). Since \s^{-3}\ is a negative exponent, it is equivalent to \(\frac{1}{s^3}\).

Putting it together, you get \(\frac{r^3}{s^3}\).

Square Roots

Taking square roots involves simplifying the expression under the radical (√). Let's simplify \(\frac{25 r^3}{64 s^3} \). The square root of a fraction is the square root of the numerator divided by the square root of the denominator:

• The square root of 25 is 5, and the square root of 64 is 8.
• For the variables: the square root of \( r^3 = r^{3/2}\) and the square root of \( s^3 = s^{3/2}\).

Thus, \(\frac{\text{sqrt} {25 r^3}}{\text{sqrt} {64 s^3}}\) simplifies to \(\frac{5 r^{3/2}}{8 s^{3/2}}\).

Numerical Coefficients

Numerical coefficients are the numbers multiplying the variables in terms. In our simplified expression \(\frac{5 r^{3/2}}{8 s^{3/2}} \), the coefficients are 5 (numerator) and 8 (denominator). Make sure to handle these coefficients separately from the variables when simplifying. Here, they were factored out during the square root process, and the final answer was obtained. Keeping the numerical coefficients separate ensures clarity and precision in mathematical operations.