Problem 91 - Algebra 1: Concepts and Skills

Question

MULTIPLE CHOICE Which step would you use to rationalize the denominator of \(\frac{\sqrt{3}}{\sqrt{10}} ?\) (F) Multiply by \(\frac{\sqrt{10}}{\sqrt{10}}\) (G) Multiply by \(\frac{\sqrt{10}}{\sqrt{3}}\) (H) Multiply by \(\sqrt{10}\) (J) Multiply by 10.

Step-by-Step Solution

Answer

The correct choice is (F) Multiply by \(\frac{\sqrt{10}}{\sqrt{10}}\).

Step 1: Identify Fraction to Rationalize

We begin with the fraction \(\frac{\sqrt{3}}{\sqrt{10}}\). We wish to rationalize the denominator (bottom) \(\sqrt{10}\).

Step 2: Decide what to Multiply

We then realize that we can eliminate the square root in the denominator by multiplying the fraction by a form of 1 that includes \(\sqrt{10}\). This boils down to \(\frac{\sqrt{10}}{\sqrt{10}}\), because any number divided by itself is 1.

Step 3: Multiply the Fraction

So now we multiply our fraction by \(\frac{\sqrt{10}}{\sqrt{10}}\). The resulting fraction is \(\frac{\sqrt{3} \times \sqrt{10}}{\sqrt{10} \times \sqrt{10}}\). This simplifies to \(\frac{\sqrt{30}}{10}\). Now the denominator is a rational number, so the denominator has been rationalized.

Key Concepts

Square Roots

Understanding square roots is fundamental when dealing with rationalizing denominators. A square root, specified as \(\sqrt{\cdot} \), of a number is a value that, when multiplied by itself, gives the original number. The number under the square root is often called the "radicand". For example, in \(\sqrt{9} \), 9 is the radicand and the square root is 3 because \(3 \times 3 = 9\). Learning how to manipulate square roots is crucial for simplifying and solving radical expressions.

When you see a fraction like \(\frac{\sqrt{3}}{\sqrt{10}}\), each part of the fraction can be individually evaluated through its square root. Breaking down and understanding each component allows for a more seamless transformation and simplification later on.

Simplifying Radicals

The process of simplifying radicals involves reducing the expression to its simplest form. This is often done by identifying perfect squares within a radical and taking them out. For instance, \(\sqrt{20}\) can be written as \(\sqrt{4 \times 5}\), which simplifies to \(2\sqrt{5}\) because \(\sqrt{4} = 2\).

When we simplify radicals in the context of rationalizing the denominator, the primary goal is to eliminate any square roots from the denominator. For example, \(\frac{\sqrt{3} \times \sqrt{10}}{\sqrt{10} \times \sqrt{10}}\) simplifies to \(\frac{\sqrt{30}}{10}\). Here, the denominator \(\sqrt{10} \times \sqrt{10}\) becomes 10, a rational number, thereby accomplishing the goal of rationalizing.

Multiplication of Fractions

Multiplication of fractions is a straightforward process but crucial in rationalizing denominators and simplifying complex expressions. To multiply fractions, multiply the numerators (tops) together and the denominators (bottoms) together. In our example, we multiplied \(\frac{\sqrt{3}}{\sqrt{10}}\) by \(\frac{\sqrt{10}}{\sqrt{10}}\).

This gives us \(\frac{\sqrt{3} \times \sqrt{10}}{\sqrt{10} \times \sqrt{10}}\). Multiplying radicals together combines their radicands under a single radical: \(\sqrt{3 \times 10} = \sqrt{30}\). The denominator becomes \(10\), since \(\sqrt{10} \times \sqrt{10} = 10\). By doing so, we've made the fraction more manageable and easier to work with.