Problem 13 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$15 \sqrt{9}$$

Step-by-Step Solution

Answer

The simplified expression is 45.

Step 1: Identify the square root

First, identify the square root in the expression. In this case, the number under the square root is 9.

Step 2: Calculate the square root

Next, we calculate the square root of 9. Since 9 is a perfect square, its square root is 3.

Step 3: Multiply the square root by the coefficient

Finally, multiply the result of the square root by the coefficient outside the square root. In this case, multiply 3 (the result from the previous step) by 15.

Step 4: Simplify the expression

The final step is to perform the multiplication: \[ 15 imes 3 = 45 \]

Key Concepts

Understanding Square Roots

Square roots are fundamental in mathematics, especially when simplifying expressions. A square root of a number is a value that, when multiplied by itself, gives the original number.
For example, \( \sqrt{9} \) means finding a number which, when squared (multiplied by itself), equals 9. This number is 3, since \( 3 \times 3 = 9 \). This concept is crucial because it often allows us to simplify expressions in prealgebra without a calculator.
Whenever you see a square root, determine if the number inside is a perfect square, making it easier to simplify. Recognize that some numbers, like 4, 9, 16, and 25, are perfect squares. They lead to whole number square roots, simplifying your calculations greatly.

Exploring Perfect Squares

Perfect squares are numbers that are the result of an integer multiplied by itself. They play a vital role in simplifying square roots and other algebraic expressions.
For instance, as in our example, 9 is a perfect square because \( 3 \times 3 = 9 \). Knowing perfect squares up to a reasonable number, like 100, can speed up calculations:

• 1, since \( 1 \times 1 = 1 \)
• 4, since \( 2 \times 2 = 4 \)
• 9, since \( 3 \times 3 = 9 \)
• 16, since \( 4 \times 4 = 16 \)
• and so on...

These numbers allow for immediate simplification of radical expressions because you know their square roots by heart. Practicing perfect squares frequently will make them second nature and improve your prealgebra skills tremendously.

Mastering Multiplication in Simplification

In prealgebra, multiplication often follows the simplification of square roots, as seen in our exercise. When you have a term outside the square root, multiplying comes next after evaluating the square root.
For example, consider \( 15 \times \sqrt{9} \). First, simplify \( \sqrt{9} \) to 3, then multiply these two values together:

• 15 (outside coefficient) \( \times \) 3 (simplified square root) gives you 45.

This process of multiplying after simplification minimizes errors and allows you to keep your calculations clear and logical. Using multiplication fluently is key in making math expressions manageable and accurate.