Problem 24 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$7 \sqrt{49}-2 \sqrt{4}$$

Step-by-Step Solution

Answer

The simplified expression is 45.

Step 1: Simplify Square Roots

First, simplify the square roots in this expression. Given: \(\sqrt{49}\) and \(\sqrt{4}\). \(\sqrt{49} = 7\) because 7 multiplied by itself is 49, and \(\sqrt{4} = 2\) because 2 multiplied by itself is 4.

Step 2: Multiply Numerical Coefficients

Now, simplify the expression by multiplying the numerical coefficients: \(7 \cdot \sqrt{49} = 7 \cdot 7 = 49\) and \(2 \cdot \sqrt{4} = 2 \cdot 2 = 4\).

Step 3: Subtract the Results

Subtract the results obtained in Step 2: \(49 - 4 = 45\).

Key Concepts

Square Roots

Square roots are numbers that, when multiplied by themselves, give you the original number. For example, \( \sqrt{49} \) means "what number times itself equals 49?" The answer is 7 because \( 7 \times 7 = 49 \). Similarly, \( \sqrt{4} \) is 2 because \( 2 \times 2 = 4 \).
Understanding square roots is important when simplifying expressions. Think of it like finding the hidden number inside the square. If you have a perfect square, like 49 or 4, its square root will be a whole number. This makes calculations easier, especially when you don't have a calculator handy.
When simplifying expressions, always start by dealing with any square roots first, so everything is in its simplest form upfront.

Numerical Coefficients

Numerical coefficients are simply the numbers in front of the square root sign. If we have expressions like \( 7 \sqrt{49} \) or \( 2 \sqrt{4} \), the numbers 7 and 2 are the coefficients. They represent how many times you are multiplying the square root value.
To simplify, multiply the coefficient by the value from the square root. For example, in \( 7 \sqrt{49} \), first calculate \( \sqrt{49} \) which is 7, and then multiply it by the coefficient 7. So, you do \( 7 \times 7 \) to get 49.
Understanding coefficients helps you manage expressions and make correctly simplified calculations.

Subtracting Results

Once you have dealt with square roots and coefficients, the next step is subtracting the results to simplify the entire expression. Subtraction is simply taking one value away from another.
In the expression \( 49 - 4 \), subtract 4 from 49. You'll get a result of 45 which is the simplest form of our original expression \( 7 \sqrt{49} - 2 \sqrt{4} \).
It's important to align the numbers correctly under subtraction to avoid mistakes. This makes the process clearer and ensures the accuracy of your final result.