Problem 47 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ \sqrt{7}(\sqrt{5}+\sqrt{3}) $$

Step-by-Step Solution

Answer

The simplified expression is \(\sqrt{35} + \sqrt{21}\).

Step 1: Distribute the Square Root

To start multiplying, distribute the \(\sqrt{7}\) to each term inside the parentheses. This means multiplying \(\sqrt{7}\) with \(\sqrt{5}\) and \(\sqrt{7}\) with \(\sqrt{3}\).

Step 2: Multiply the Square Roots

Combine \(\sqrt{7}\) and \(\sqrt{5}\) by using the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Thus, \(\sqrt{7} \cdot \sqrt{5} = \sqrt{35}\) and \(\sqrt{7} \cdot \sqrt{3} = \sqrt{21}\).

Step 3: Write the Combined Expression

After multiplying, the expression becomes \(\sqrt{35} + \sqrt{21}\). This is already simplified because there are no perfect square factors in \(35\) or \(21\) that we can extract.

Key Concepts

Properties of Square Roots

Understanding the properties of square roots is important when working with radicals. The square root function has a few distinctive properties that make calculations easier and allow for simplification of expressions.

• Product Property: This states that the square root of a product is equal to the product of the square roots of the individual factors. Mathematically, \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). This property is crucial when multiplying square root terms together as it provides a direct way to combine them.
• Quotient Property: Similarly, the square root of a quotient is the quotient of the square roots of the numerator and denominator: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This is useful for simplifying fractions under a radical.

These properties often make apparent the relationship between numbers, allowing further simplification and manipulation of expressions. Keep them in mind as they frequently apply to radical expressions in different contexts.

Distributing Square Roots

When multiplying, distribution is an essential concept that extends to square roots as well. Distributing helps to manage terms when multiplying a single radical across a sum or a difference.

• Distribution Steps: Let's consider the expression \( \sqrt{7}(\sqrt{5}+\sqrt{3}) \). Here, you distribute the \( \sqrt{7} \) by multiplying it with each term inside the parenthesis separately. So, you perform these operations: \( \sqrt{7} \times \sqrt{5} \) and \( \sqrt{7} \times \sqrt{3} \).
• Resulting Terms: Using the product property of square roots from before, we get: \( \sqrt{35} \) from the first multiplication and \( \sqrt{21} \) from the second. This distribution ensures that every term in the parenthesis is accounted for and related back to the single term outside it.

Distribution helps in breaking down and eventually simplifying polynomial-like expressions, allowing for manageable calculations with radicals.

Simplification of Radicals

Once multiplication and distribution are handled, it is often necessary to simplify the resulting radicals. Simplification aims to express the square root in its most reduced form.

• Finding Perfect Squares: Check if the number under the square root has factors that are perfect squares. If it does, you can simplify the radical by taking the square root of the factor and bringing it outside the radical sign.
• Example Application: In our expression \( \sqrt{35} + \sqrt{21} \), neither 35 nor 21 have perfect square factors other than 1. Therefore, these radicals are in their simplest form and any further simplification isn't possible. If they had factors like 4 or 9, we could have extracted a factor to simplify further.

Thus, while not always possible, finding and using perfect squares that divide inside the radical allows for the simplest expression. Noticing when a radical is already at its simplest can save time and effort in calculations.