Problem 16 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$4 \sqrt{50}-5 \sqrt{8}$$

Step-by-Step Solution

Answer

The simplified expression is \( 10 \sqrt{2} \).

Step 1: Simplify each square root term

First, simplify each square root term separately. We begin with \( \sqrt{50} \). We can write \( 50 \) as \( 25 \times 2 \), so \( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2} \). Next, simplify \( \sqrt{8} \) by rewriting it as \( 4 \times 2 \), so \( \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2 \sqrt{2} \).

Step 2: Substitute the simplified square roots back into the expression

Substitute the simplified expressions back into the original expression: \( 4 \sqrt{50} - 5 \sqrt{8} = 4 \cdot (5 \sqrt{2}) - 5 \cdot (2 \sqrt{2}) \).

Step 3: Calculate each term by multiplication

Multiply the coefficients: \( 4 \cdot 5 \sqrt{2} = 20 \sqrt{2} \) and \( 5 \cdot 2 \sqrt{2} = 10 \sqrt{2} \). The expression is now \( 20 \sqrt{2} - 10 \sqrt{2} \).

Step 4: Combine like terms

Since both terms share the same radical \( \sqrt{2} \), subtract the coefficients: \( 20 - 10 = 10 \). Thus, the expression simplifies to \( 10 \sqrt{2} \).

Key Concepts

Understanding Square Roots

A square root finds a number which, when multiplied by itself, results in the original number. For instance, the square root of 25 is 5 because 5 multiplied by 5 equals 25. In algebra, simplifying square roots often involves breaking down the number inside the square root into its factors.
For example, to simplify \( \sqrt{50} \), consider the prime factors: 50 can be broken down into \( 25 \times 2 \). The square root of 25 is a perfect square, which simplifies further:

• \( \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} \)
• This becomes \( 5\sqrt{2} \), as \( \sqrt{25} \) equals 5.

This process of splitting numbers into products of squares assists in simplifying many expressions with square roots. Remember this strategy as it's a valuable tool for simplification.

Combining Like Terms in Expressions

Combining like terms simplifies expressions by merging terms that have identical variables or square root parts. To determine if terms are alike, look at the numbers under the square root rather than outside coefficients.
In the expression \( 4 \sqrt{50} - 5 \sqrt{8} \), both can be reduced to forms with like radicals:

• \( 4 \sqrt{50} \) simplifies to \( 20 \sqrt{2} \)
• \( 5 \sqrt{8} \) simplifies to \( 10 \sqrt{2} \)

After simplification:

• Both terms share the same \( \sqrt{2} \), allowing them to be combined.

Subtract the coefficients of these like terms, as demonstrated in combining \( 20 \sqrt{2} \) and \( -10 \sqrt{2} \), resulting in \( 10 \sqrt{2} \).

Coefficients Multiplication with Radicals

When multiplying expressions with square roots, you multiply the coefficients (the numbers outside the square root) separately from the numbers inside the square root.
Consider the initial expression \( 4 \sqrt{50} - 5 \sqrt{8} \). After simplifying the square roots, you have:

• \( 4 (5 \sqrt{2}) \) turning into \( 20 \sqrt{2} \)
• \( 5 (2 \sqrt{2}) \) turning into \( 10 \sqrt{2} \)

Here's how the multiplication was done:

• The coefficients 4 and 5 became 20 as a result of multiplying 4 by the simplified square root coefficient 5.
• Similarly, for \( 5\cdot (2 \sqrt{2}) \), the coefficients 5 and 2 became 10.

The essence of coefficients multiplication lies in handling numbers outside the square root, while the like terms are found based upon the square root itself.