Rationalize the denominator.17\frac{1}{\sqrt7}71

Ch. P - Fundamental Concepts of Algebra

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Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

문제 45

1장, 문제 45

Rationalize the denominator.
$\frac{1}{\sqrt7}$

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Identify the expression to rationalize: $\frac{1}{\sqrt7}$, where the denominator contains a square root.

To rationalize the denominator, multiply both the numerator and the denominator by $\sqrt7$ to eliminate the square root in the denominator.

Write the multiplication as: $\frac{1}{\sqrt7} \times \frac{\sqrt7}{\sqrt7}$.

Multiply the numerators together and the denominators together: numerator becomes $1 \times \sqrt7 = \sqrt7$, denominator becomes $\sqrt7 \times \sqrt7 = 7$.

Rewrite the expression with the rationalized denominator as $\frac{\sqrt7}{7}$.

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Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers, such as square roots, from the denominator of a fraction. This is done by multiplying the numerator and denominator by a suitable expression that removes the radical, making the expression easier to interpret and use in further calculations.

Properties of Square Roots

Square roots represent a number which, when multiplied by itself, gives the original number. Understanding that √a × √a = a is essential for rationalizing denominators, as multiplying by the square root in the denominator can simplify the expression by removing the radical.

Multiplying Fractions by 1

Multiplying a fraction by a form of 1, such as √7/√7, does not change its value but can change its form. This technique is used to rationalize denominators by creating an equivalent fraction with a rational denominator, facilitating easier manipulation and simplification.

Rationalize the denominator.17\(\frac{1}{\sqrt7}\)71