Problem 134 - Elementary Algebra

Question

Simplify. $$ \sqrt{\frac{49 r^{12}}{16 r^{6}}} $$

Step-by-Step Solution

Answer

\( \frac{7 r^3}{4} \)

Step 1: Simplify the Fraction Inside the Square Root

First, simplify the expression inside the square root. The original fraction is \( \frac{49 r^{12}}{16 r^{6}} \). Separate the constants and the variables: \( \frac{49}{16} \) and \( \frac{r^{12}}{r^{6}} \).

Step 2: Simplify the Constants

The fraction \( \frac{49}{16} \) is already in its simplest form, since 49 and 16 have no common factors.

Step 3: Simplify the Variables

For the variables, use the property \( \frac{a^m}{a^n} = a^{m-n} \). Therefore, \( \frac{r^{12}}{r^{6}} = r^{12-6} = r^6 \).

Step 4: Combine Simplified Parts

Now, combine the simplified parts back together: \[ \sqrt{\frac{49 r^{12}}{16 r^{6}}} = \sqrt{\frac{49}{16} \cdot r^6} = \sqrt{\frac{49}{16}} \cdot \sqrt{r^6} \].

Step 5: Simplify the Square Roots

Simplify each square root separately. \( \sqrt{\frac{49}{16}} = \frac{\sqrt{49}}{\sqrt{16}} = \frac{7}{4} \), and \( \sqrt{r^6} = r^{\frac{6}{2}} = r^3 \).

Step 6: Multiply the Simplified Parts

Multiply the simplified parts together: \[ \frac{7}{4} \cdot r^3 = \frac{7 r^3}{4} \].

Key Concepts

fraction simplification

To simplify fractions, you need to reduce the fraction to its simplest form. This means you want the numerator (top number) and the denominator (bottom number) to have no common factors other than 1. In the exercise, the fraction is \(\frac{49}{16}\). 49 and 16 have no common factors besides 1, so the fraction is already simplified. Always look to break down the numbers into their prime factors and cancel any common factors. This process ensures the fraction is as simple as possible.
Key steps to remember:

• Find common factors.
• Divide both the numerator and denominator by their greatest common divisor (GCD).

Practice simplifying different fractions to get familiar with the process. Simplifying fractions helps make complex problems easier to solve.

variable exponents

Variable exponents can appear tricky, but they're manageable if you know the rules. The key principle to remember is \[ \frac{a^m}{a^n} = a^{m-n} \]. For instance, in \( \frac{r^{12}}{r^{6}} \), you subtract the exponent in the denominator from the exponent in the numerator: \ r^{12-6} = r^6 \ . This rule is crucial for simplifying terms with similar bases.
Be careful to apply it only when both the base and the variable are the same.
To master this, continuously practice:

• Identifying similar bases.
• Subtracting exponents correctly.
• Applying the rule to different problems.

Understanding this concept will make it much easier to handle complex algebraic expressions.

square roots

Square roots involve finding a number which, when multiplied by itself, gives the original number. For an algebraic expression, separate the constants and variables. For example,
\[ \sqrt{\frac{49}{16}} = \frac{\sqrt{49}}{\sqrt{16}} = \frac{7}{4} \] \ and \ \sqrt{r^6} = r^{6/2} = r^3 \
Each term is simplified separately and then combined. Here are some key steps:

• Take the square root of each part of the expression individually.
• Simplify constants by recognizing perfect squares.
• Apply the property \[ \sqrt{a^b} = a^{b/2} \] to variables.

Practice these steps repetitively to get comfortable with simplifying square roots in various algebraic contexts.