Problem 82 - Algebra and Trigonometry

Question

Exercises 81–83 will help you prepare for the material covered in the next section. Simplify and round to the nearest whole number: $$ \sqrt{26(26-12)(26-16)(26-24)} $$

Step-by-Step Solution

Answer

The answer to the expression \( \sqrt{26(26-12)(26-16)(26-24)} \) is 302 when rounded to the nearest whole number.

Step 1: Simplify inside the square root

Start by simplifying the expression inside the square root. This involves carrying out the operations in parentheses: \(26-12\) results in 14, \(26-16\) gives 10 and \(26-24\) gives 2. Then multiply these by 26. This simplifies the expression to \( \sqrt{26 \times 14 \times 10 \times 2} \)

Step 2: Calculate the expression under the square root

Next, perform the multiplication under the square root. \(26\times14\times10\times2 = 91280\). The expression now simplifies to \(\sqrt{91280}\).

Step 3: Find the square root

The next step is to compute the square root of 91280. This results in approximately 302.12133.

Step 4: Round to the nearest whole number

Finally, round the result to the nearest whole number. Rounded to nearest whole number, the result is 302.

Key Concepts

Multiplication

Multiplication is a fundamental arithmetic operation that involves combining groups of equal sizes. It's a shortcut for repeated addition. For example, instead of adding 14 three times (14 + 14 + 14), you can multiply 14 by 3 to get 42.
In algebraic expressions, multiplication allows for the simplification of terms. When you have \(a imes b\), you are effectively creating a single term with a value equal to the product of a and b. In the context of our problem, after breaking down the expression into 26, 14, 10, and 2, each is multiplied together to find a single value under the square root.
Using multiplication in algebra helps to reduce complex fractions, like in the expression \(26 \times 14 \times 10 \times 2\), which made it easier to handle large numbers and to subsequently find the square root efficiently.

Rounding Numbers

Rounding numbers is a method of adjusting figures to make them simpler and often easier to use. This method is particularly helpful when you want to convey information without a lot of unnecessary precision.
For numbers, rounding means finding the nearest significant value. Depending on context, you might round to the nearest tens, hundreds, or to a specific decimal place. In our example from the solution, we rounded to the nearest whole number.

• Look at the digit right after where you're rounding. If it’s five or more, round up.
• If it’s less than five, keep the digit the same.

For \( ext{302.12133}\), we look at the tenths place (1), which tells us we should round 302 down to just 302. This simplification makes the number more manageable and user-friendly in practical applications.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can involve numbers, variables, like x or y, and operations like addition or multiplication. They are the building blocks of algebra, allowing you to solve equations and understand relationships between quantities.
In our exercise, \(26(26-12)(26-16)(26-24)\) is an algebraic expression involving both numbers and operations. It condenses complex operations into a manageable form. Each number in parentheses represents a transformation of 26 followed by multiplication.
Understanding how to expand and simplify algebraic expressions is fundamental in solving algebra problems. Simplifying involves performing operations in the correct order. This often includes breaking down the expression into smaller parts, solving each part, and then combining for the final expression, as we did to reach \( ext{91280}\) inside the square root.
Algebraic expressions help in visualizing and problem-solving across many topics, including finance, engineering, and even in the square root simplification problem presented.