Problem 23 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$15 \sqrt{9}-9 \sqrt{16}$$

Step-by-Step Solution

Answer

The expression simplifies to 9.

Step 1: Simplify Each Square Root

First, simplify each square root. The expression contains \(\sqrt{9}\) and \(\sqrt{16}\). We know from basic square root properties that \(\sqrt{9} = 3\) and \(\sqrt{16} = 4\).

Step 2: Substitute the Simplified Values

Replace \(\sqrt{9}\) with \(3\) and \(\sqrt{16}\) with \(4\) in the original expression. This gives us \(15 \times 3 - 9 \times 4\).

Step 3: Perform the Multiplications

Calculate the products: \(15 \times 3 = 45\) and \(9 \times 4 = 36\).

Step 4: Solve the Expression

Subtract the two results obtained in Step 3. This is \(45 - 36 = 9\).

Key Concepts

Square Roots

Understanding square roots is crucial for simplifying expressions like the one in our original exercise. A square root essentially "un-dos" a square. For instance, if you have a number like 9, and you know that 9 is the result of squaring 3 (3^2 = 9), then \(\sqrt{9}\) brings you back to the original number, 3. It is the non-negative number that, when multiplied by itself, gives the original value.

Let's break it down further with our example from the original expression. We have \sqrt{9}\, and since 3 times 3 gives 9, that means \sqrt{9} = 3\. Similarly, \sqrt{16} = 4\ because 4 multiplied by 4 equals 16.

To simplify square roots effectively:

• Identify the perfect square that matches your square root (like 9, 16, 25, etc.).
• Find the number that when squared gives you the original number under the root.
• Use this number to replace the square root in your expression.

Multiplication

Multiplication is a mathematical operation involving the process of combining quantities. It's a means of calculating the total of one number found so many times. In our expression, we used multiplication to calculate parts of the expression separately.

In the expression part \(15 \times \sqrt{9}\), we simplified \ \sqrt{9} \ to 3, and then multiplied it by 15. Doing this gives us \(15 \times 3 = 45\). Similarly, we simplified \ \sqrt{16} \ to 4 and multiplied it by 9, which results in \(9 \times 4 = 36\).

Here are some multiplication basics that can help simplify calculations:

• Multiplying by 1 keeps the number the same.
• Multiplying by 0 always results in 0.
• Products do not change even if the numbers' order is reversed (commutative property), so \(a \times b = b \times a\).

Subtraction

Subtraction might seem straightforward, but it’s important when simplifying expressions. Subtraction helps determine the difference between quantities, which in simpler terms, means removing a number from another number.

After performing multiplications in our original expression, we were left with two results: 45 and 36. The next logical step is to subtract the smaller product from the larger one. So, \(45 - 36 = 9\). That's how we simplified the whole expression to just 9 by using subtraction.

To ensure accuracy when subtracting, keep the following tips in mind:

• Subtraction is not commutative, meaning \(a - b eq b - a\).
• Always subtract the smaller number from the larger one to avoid negative results unless intended.
• Double-check your results by reversing the process (i.e., spread and reduce to verify).