Problem 361 - Elementary Algebra

Question

In the following exercises, simplify and rationalize the denominator. $$ \sqrt{\frac{19}{175}} $$

Step-by-Step Solution

Answer

\( \frac{\root{133}}{35} \)

Step 1: Separate the Fraction

First, rewrite the expression by separating the numerator and the denominator inside the radical sign: \[\frac{\root{19}}{\root{175}}\].

Step 2: Simplify the Denominator

To simplify the denominator, factorize 175: \[175 = 25 \times 7 = 5^2 \times 7\]. This changes the expression to \[\frac{\root{19}}{\root{5^2 \times 7}} = \frac{\root{19}}{5 \root{7}}\].

Step 3: Rationalize the Denominator

To rationalize the denominator, multiply both the numerator and the denominator by \(\root{7} \) so that the denominator becomes a rational number: \[\frac{\root{19} \times \root{7}}{5 \root{7} \times \root{7}} = \frac{\root{133}}{5 \times 7} = \frac{\root{133}}{35}\].

Key Concepts

simplify radical expressions

Simplifying radical expressions involves making them easier to work with. A radical expression contains a square root, cube root, or other roots. The main goal is to break down the expression into its simplest form.

In our example, we started with \(\root{ \frac{19}{175} \). To simplify this, we rewrote it by separating the numerator and the denominator inside the radical sign. This gives us \(\frac{ \root{19} }{ \root{175} } \).

Next, we simplified the term under the root in the denominator, by factoring 175 into prime numbers. This step makes it easier to handle the expression and allows us to simplify further.

factorization

Factorization is the process of breaking down a number into smaller numbers that, when multiplied together, give the original number. This is especially useful in simplifying radical expressions.

For instance, the number 175 can be factorized as follows:

• 175 = 25 × 7
• 25 can further be broken down to 5 × 5
• Thus, 175 = 5 × 5 × 7 = 5^2 × 7

Using this factorization in our example, we get the expression \(\frac{ \root{19} }{ \root{5^2 \times 7} } \). This simplifies to \(\frac{ \root{19} }{ 5 \root{7} } \), making it easier to rationalize the denominator next.

basic algebra

Basic algebra involves working with variables, constants, and basic operations such as addition, subtraction, multiplication, and division. When dealing with radical expressions, these operations help simplify and rationalize the expressions.

Rationalizing the denominator is a common practice in algebra. In our example, we multiplied both the numerator and the denominator by \(\root{7} \), to eliminate the radical in the denominator. This step is crucial because a rationalized denominator is free of radicals and is considered a simplified form.

Multiplying \(\root{7} \) to the denominator, we have:

• \(\frac{ \root{19} \times \root{7} }{ 5 \root{7} \times \root{7} } = \frac{ \root{133} }{ 5 \times 7 } \)
• The result is \(\frac{ \root{133} }{ 35 } \), which is in its simplest form.