Problem 48 - Algebra 1

Question

Simplify the expression. $$\frac{\sqrt{10} \cdot \sqrt{16}}{\sqrt{5}}$$

Step-by-Step Solution

Answer

\(\sqrt{32}\)

Step 1: Separate the square roots

The first step is to apply the multiplication rule for square roots, which allows you to separate \(\sqrt{10} \cdot \sqrt{16}\) into \(\sqrt{10 \cdot 16}\) .

Step 2: Simplify the Numerator

After the separation, you calculate \(10 \cdot 16\) which results to \(160\), thus the expression becomes \(\sqrt{160}/\sqrt{5}\).

Step 3: Apply the rule to the denominator

Next, you apply the rule for division inside a square root which says the square root of a quotient is the quotient of square roots. Therefore, \(\sqrt{160}/\sqrt{5}\) could be expressed as \(\sqrt{160/5}\).

Step 4: Simplify and calculate the remaining expression

Finally, calculate \(160/5\) which is \(32\), and take the square root of \(32\), which equals to \(\sqrt{32}\).

Key Concepts

Square Roots

The term "square root" refers to a value that, when multiplied by itself, gives the original number. For any positive number, the square root is denoted by the radical symbol \( \sqrt{} \). For example, \( \sqrt{16} = 4 \) because \( 4 \times 4 = 16 \).

Working with square roots often involves understanding the properties of radicals and using these properties to simplify expressions:

• The square root of a product can be expressed as the product of the square roots. This means for two positive numbers \( a \) and \( b \), \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \).
• Similarly, the square root of a quotient can be written as the quotient of their square roots: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) when \( b eq 0 \).

These rules help break down complex expressions into simpler components.

Multiplication of Radicals

When multiplying radicals, the radicals must have the same type of root, usually a square root. The multiplication of radicals involves using the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).

Let's look at the example from the original exercise: \( \sqrt{10} \cdot \sqrt{16} \). Using our rule, we can combine these under one radical: \( \sqrt{10 \times 16} = \sqrt{160} \). This simplification is helpful as it often leads to an expression that can be further simplified.

Always remember to multiply the values under the radicals first before attempting to simplify further. This can make the process much quicker and help avoid mistakes.

Division of Radicals

Dividing radicals can initially seem tricky, but using rules for radicals makes it straightforward. When you divide two square roots, such as \( \frac{\sqrt{a}}{\sqrt{b}} \), you can express it as a single square root: \( \sqrt{\frac{a}{b}} \), provided \( b eq 0 \).

In our exercise, we simplified the expression \( \frac{\sqrt{160}}{\sqrt{5}} \) using this rule. This becomes \( \sqrt{\frac{160}{5}} \) which simplifies to \( \sqrt{32} \) after dividing 160 by 5.

Using this rule often simplifies expressions significantly, reducing the complexity and making further simplification possible. Be cautious with negative numbers as they can introduce complexities like imaginary numbers.

Simplification Techniques

Simplification techniques are essential for breaking down complex radical expressions into more manageable ones. Once you have a single radical, like \( \sqrt{32} \) from the original problem, check if it can be simplified further.

Identify any perfect square factors in the number under the square root. \( 32 \) can be broken down into \( 16 \times 2 \). Since \( \sqrt{16} = 4 \), \( \sqrt{32} \) simplifies to \( 4\sqrt{2} \).

Here are some quick tips:

• Look for factors that are perfect squares to simplify the expression easily.
• Break the number under the radical into these factors.
• Simplify each part where possible, allowing for cleaner, simpler final expressions.

This approach makes the radical operations less overwhelming and more intuitive, leading to clearer results.