Question

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{-\sqrt{3}}{\sqrt{5}}$$

Short answer

\(\frac{-\text{√}15}{5}\)

Step-by-step solution

Identify the denominator

The given expression is \(\frac{-\text{√}3}{\text{√}5}\). Here, the denominator is \(\text{√}5\).

Multiply numerator and denominator by the conjugate of the denominator

To rationalize the denominator, multiply both the numerator and the denominator by \(\text{√}5\). The expression becomes \(\frac{-\text{√}3 \times \text{√}5}{\text{√}5 \times \text{√}5}\).

Simplify the expression

Simplify the numerator and the denominator. The numerator \(-\text{√}3 \times \text{√}5\) simplifies to \(-\text{√}(3 \times 5) = -\text{√}15\). The denominator \(\text{√}5 \times \text{√}5\) simplifies to \(\text{5}\). Thus, the rationalized form is \(\frac{-\text{√}15}{5}\).

Key concepts

rational expression

A rational expression is a fraction where the numerator and/or the denominator are polynomials. Understanding rational expressions is crucial in algebra.
In our exercise, we started with an expression that had a radical (square root) in the denominator. When dealing with rational expressions, having a radical in the denominator is not considered simplified.
Here's why rationalizing the denominator is important:

• It simplifies the arithmetic operations.
• It makes expressions easier to interpret and compare.
• It is often required in standardized mathematical formats.

To rationalize the denominator, we typically multiply both the numerator and the denominator by some form that will eliminate the radical in the denominator.

simplifying radicals

Simplifying radicals involves reducing them to their most simplified form. This makes mathematical expressions easier to work with.
In our exercise, we began with: \(\frac{-\text{√}3}{\text{√}5}\). Simplifying radicals means dealing with the square roots directly.
Here’s what we did step-by-step:

• Identify and rewrite the expression:

Starting with \(\frac{-\text{√}3}{\text{√}5}\).

Multiply numerator and denominator by \(\text{√}5\):

The result becomes \(\frac{-\text{√}3 \times \text{√}5}{\text{√}5 \times \text{√}5}\).

Simplify both numerator and the denominator:

The numerator simplifies to \(-\text{√}(3 \times 5) = -\text{√}15\). The denominator becomes 5 since \(\text{√}5 \times \text{√}5 = 5\).Finally, we have the simplified expression: \(\frac{-\text{√}15}{5}\).

conjugate

The conjugate of a binomial expression is formed by changing the sign between two terms. When rationalizing denominators containing radicals, we often use the conjugate to eliminate the radicals.
In this exercise, our denominator was \(\text{√}5\). To rationalize this, multiplying by the conjugate would technically involve changing a sign, but since this is purely a square root, simply multiplying by \(\text{√}5\) works similarly.
The concept of a conjugate is more commonly used when dealing with binomials (two-term expressions) such as \(\text{√}a + b\). Its conjugate would be \(\text{√}a - b\). Using the conjugate helps us eliminate radicals in denominators and is essential in making rational expressions more manageable.
Remember, multiplying by the conjugate helps get rid of the square root in the denominator through difference of squares.