Problem 17 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$2 \sqrt{90}+3 \sqrt{40}-4 \sqrt{10}$$

Step-by-Step Solution

Answer

The expression simplifies to \(8\sqrt{10}\).

Step 1: Simplify Each Square Root

First, simplify each square root. Start with \( \sqrt{90} \). Factor 90 as \( 9 \times 10 \), where 9 is a perfect square. Hence, \( \sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \cdot \sqrt{10} = 3 \cdot \sqrt{10} \).Next, simplify \( \sqrt{40} \). Factor 40 as \( 4 \times 10 \), where 4 is a perfect square. Hence, \( \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \cdot \sqrt{10} = 2 \cdot \sqrt{10} \).\( \sqrt{10} \) is already simplified.

Step 2: Substitute Simplified Roots

Substitute the simplified square roots back into the expression:\[2 \sqrt{90} + 3 \sqrt{40} - 4 \sqrt{10} = 2(3\sqrt{10}) + 3(2\sqrt{10}) - 4\sqrt{10}\]This simplifies to:\[6\sqrt{10} + 6\sqrt{10} - 4\sqrt{10}\]

Step 3: Combine Like Terms

Now combine the terms that have \( \sqrt{10} \):\[6\sqrt{10} + 6\sqrt{10} - 4\sqrt{10} = (6 + 6 - 4)\sqrt{10} = 8\sqrt{10}\]

Step 4: Final Simplified Expression

The expression simplifies to \( 8\sqrt{10} \). This is the simplified form of the given expression where all possible like terms have been combined.

Key Concepts

Combining Like Terms

When working with expressions that involve square roots, combining like terms is an essential skill. In the expression given, each term initially includes a multiple of a common square root, that is, \( \sqrt{10} \). Notice that the simplified forms of \( \sqrt{90} \) and \( \sqrt{40} \) both also involve \( \sqrt{10} \):

• \( 6\sqrt{10} \)
• \( 6\sqrt{10} \)
• \( -4\sqrt{10} \)

To combine like terms, add or subtract the coefficients (the numbers in front) of the same square root. Think of it as simply adding together the numbers that count the same kind of object:

• Adding \( 6 + 6 = 12 \)
• Subtracting \( 4 \) from \( 12 \) gives \( 8 \)

Finally, you get \( 8\sqrt{10} \), where 8 is the result from combining like terms.

Perfect Squares

Perfect squares play a fundamental role when simplifying square roots. Recognizing a perfect square helps to reduce a square root expression more simply. A perfect square is any integer that can be written as another integer squared. For example:

• \( 4 = 2^2 \)
• \( 9 = 3^2 \)
• \( 16 = 4^2 \)

In the context of our problem, numbers like 9 and 4 were identified as perfect squares, which made simplifying \( \sqrt{90} \) and \( \sqrt{40} \) straightforward.

• For \( \sqrt{90} \), factor into \( 9 \times 10 \), which simplifies due to 9 being a perfect square.
• Similarly, for \( \sqrt{40} \), factor into \( 4 \times 10 \).

Using perfect squares helps to simplify expressions by removing those squares from the square root, resulting in simpler terms that can be more easily handled in calculations.

Factorization

Factorization breaks down a number into its factors, the building blocks of numbers. It is useful not only in arithmetic and algebra but also in simplifying square roots. The first step in simplifying a square root often involves factorizing the number within that root:

• Start by selecting pairs of factors for the number under the square root.
• Identify which pair contains the largest perfect square if it exists.

In the exercise:

• 90 was broken down into \( 9 \times 10 \) with \( 9 \) being a perfect square, aiding in simplification.
• Similarly, 40 was split into \( 4 \times 10 \). Here \( 4 \) is a perfect square.

The process of factorization allows one to reveal hidden perfect squares, enabling simplification. Practicing factorization speeds up this process and makes finding and using these tools second nature in larger or more complicated expressions.