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Chapter 10: Problem 42

Rationalize each denominator. Assume that all variables represent positive numbers. $$ \sqrt{\frac{6}{7}} $$

Short Answer

Expert verified
\(\frac{\sqrt{42}}{7}\)

Step by step solution

01

- Identify the Denominator

In the given expression \(\text{\( \sqrt{\frac{6}{7}} \)}\), the denominator is 7.
02

鈥� Multiply by the Conjugate

To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt{7} \) (the square root of the denominator): \[ \sqrt{\frac{6}{7}} \times \ \frac{\sqrt{7}}{\sqrt{7}}. \]
03

鈥� Simplify the Numerator

Expand the expression in the numerator: \[ \frac{\sqrt{6} \cdot\ \sqrt{7}}{7}. \] This can be simplified to: \[ \frac{\sqrt{42}}{7}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

square roots
Square roots are mathematical expressions that find a number which, when multiplied by itself, results in the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. Square roots are often represented with the radical symbol (鈭�).
In algebra, square roots provide solutions to quadratic equations. When you encounter \(\text{鈭歛}\), it implies you are looking for a number that when squared will give you 'a'.
rationalizing fractions
Rationalizing fractions is a method used to eliminate square roots or other irrational numbers from the denominator of a fraction. This process makes calculations easier and often provides a clearer answer. Let's break down the steps:
  • Step 1: Identify the denominator that's irrational.
  • Step 2: Multiply both the numerator and the denominator by a value that will remove the square root from the denominator. Usually, this means multiplying by the square root of the denominator itself.
  • Step 3: Simplify the resulting expression.

For example, with the fraction \(\text{鈭歿6/7}}\), the denominator is 7. To rationalize it, you multiply both the numerator and the denominator by \(\text{鈭�7}\). This turns the expression into \(\text{鈭�(6脳7)/7}\), which simplifies to \(\text{鈭�42/7}\).
simplifying algebraic expressions
Simplifying algebraic expressions involves using various algebra rules to combine like terms and reduce the expression to its simplest form. Here鈥檚 how to approach it:
  • Combine like terms: Look for terms that have the same variable raised to the same power.
  • Use arithmetic operations: Addition, subtraction, multiplication, and division rules help simplify the expression.
  • Apply rules of exponents: Understand how to manage terms with exponents, such as \(\text{a^m脳a^n = a^{m+n}}\).

For example, in rationalizing \(\text{鈭�42/7}\), we see that the numerator is simplified to \(\text{鈭�42}\), because it's the product of \(\text{鈭�6}\) and \(\text{鈭�7}\). Then, the fraction \(\text{鈭�42/7}\) is already in its simplest form as there are no common factors to reduce further.

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