Problem 12 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$11 \sqrt{100}$$

Step-by-Step Solution

Answer

The simplified form of the expression is 110.

Step 1: Recognize the Square Root

First, identify the number inside the square root. In this case, it's 100. We need to find the square root of 100.

Step 2: Calculate the Square Root

The square root of 100 is the number that, when multiplied by itself, gives 100. This number is 10 because \( 10 \times 10 = 100 \). Hence, \( \sqrt{100} = 10 \).

Step 3: Simplify the Expression

Now that we know \( \sqrt{100} = 10 \), substitute 10 back into the expression \( 11 \sqrt{100} \), which becomes \( 11 \times 10 \).

Step 4: Multiply

Finally, perform the multiplication: \( 11 \times 10 = 110 \). This is the simplified form of the expression.

Key Concepts

Understanding Square Roots

Square roots can seem tricky, but they're actually quite simple with some practice. A square root asks the question: "What number multiplies by itself to give this value?" Square roots are represented by the symbol \( \sqrt{} \). If you see \( \sqrt{100} \), it means we need a number that, when multiplied by itself, equals 100.
For most perfect square numbers like 100, 64, or 36, their roots are integers. So, \( \sqrt{100} = 10 \) because \( 10 \times 10 = 100 \). Knowing the basic square roots of numbers like 1, 4, 9, 16, and so on, will help you simplify expressions quickly.

• Remember the perfect squares (e.g., 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, etc.)
• To find a square root, determine which integer multiplied by itself equals the given number.

Multiplication Basics

Multiplication is one of the fundamental operations in prealgebra. It's like repeated addition. For example, \( 11 \times 10 \) means adding 11 together 10 times. The important thing about multiplication is recognizing patterns and shortcuts.
Breaking numbers into smaller, more manageable parts can help. For example, to multiply 11 and 10, you can think
of 11 as \( 10 + 1 \). Using
the distributive property, you can see it as:

• \( 11 \times 10 = (10 + 1) \times 10 = 10\times 10 + 1 \times 10 = 100 + 10 = 110 \)

This shows that by breaking numbers down, you can simplify and quickly solve multiplication problems without a calculator.

Prealgebra Concepts in Simplification

Prealgebra involves basic arithmetic skills which are foundational for understanding algebra. When you simplify expressions, you apply these skills to make complex problems more manageable. In the exercise \( 11 \sqrt{100} \), we used both square roots and multiplication concepts.
The flow looks like this:

• Find the square root of 100 (which is 10).
• Replace the square root in the expression with 10.
• Multiply 11 by 10 to get 110.

Understanding how different prealgebra concepts work together can make it easier to solve math problems and build a strong foundation for more advanced topics later on. Simplifying expressions regularly will help reinforce these crucial skills.