Problem 77 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. \(\frac{1}{4} \sqrt{90}\)

Step-by-Step Solution

Answer

The simplified form of \(\frac{1}{4} \sqrt{90}\) is \(\frac{3}{4} \sqrt{10}\).

Step 1: Simplify the square root

Break down 90 inside the square root into 9 and 10. So, \( \sqrt{90} \) becomes \( \sqrt{9*10} \).

Step 2: Break down the square root into simpler terms

Re-write \( \sqrt{9*10} \) as \( \sqrt{9} * \sqrt{10} \). Therefore, \( \sqrt{90} \) simplifies to \( 3 \sqrt{10} \).

Step 3: Multiply by \(\frac{1}{4}\)

Now, multiply \(3 \sqrt{10}\) by \(\frac{1}{4}\).

Key Concepts

Simplifying Radicals

Simplifying radicals involves breaking down a complex root expression into simpler components. To do this, look for perfect square factors of the number under the radical sign (the radicand).
For example, in the expression \( \sqrt{90} \), we can simplify by finding factors of 90 that include perfect squares.
90 can be split into 9 and 10, where 9 is a perfect square because \( 3^2 = 9 \).
So we re-write \( \sqrt{90} \) as \( \sqrt{9 \times 10} \).
This step allows the radical expression to turn into something more manageable.

• Look for factors of the radicand that are perfect squares.
• Split the radicand into a product of simpler numbers.
• Re-write the radical expression using these factors for easier calculation.

This way, by simplifying radicals, you make complex expressions much easier to work with.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. It's represented by the radical symbol \( \sqrt{} \).
In this context, the square root of 9 is 3 because \( 3 \times 3 = 9 \).
When simplifying square roots, try to identify and extract perfect squares from the radicand.
For example, with \( \sqrt{90} \):

• Recognize that 9 is a perfect square.
• Extract the square root of 9, which is 3, leaving you with \( 3 \sqrt{10} \).

By finding the square root of a component, you've made the expression easier to handle.

Multiplication of Radicals

When multiplying radicals (expressions containing roots), you can apply the rule \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).
However, if you're multiplying a radical by a non-radical coefficient, it's typically done by distributing the coefficient over what’s left inside and outside of the radical.
In the final step of our example, we multiply \( 3 \sqrt{10} \) by \( \frac{1}{4} \):

• The non-radical 3 is multiplied by \( \frac{1}{4} \), resulting in \( \frac{3}{4} \).
• Thus, the simplified expression is \( \frac{3}{4} \sqrt{10} \).

Remember, when dealing with multiplication in radicals, focus on coefficients and radicands separately first, then see if they can combine further.