Problem 18 - Intermediate Algebra

Question

Add or subtract. $$\sqrt{16}-5 \sqrt{10}+7$$

Step-by-Step Solution

Answer

The expression simplifies to \( 11 - 5 \sqrt{10} \).

Step 1: Simplify the Square Root

First, we simplify the square root expression in \( \sqrt{16} \). Since 16 is a perfect square, \( \sqrt{16} = 4 \).

Step 2: Substitute the Simplified Value into the Expression

Replace \( \sqrt{16} \) with 4 in the expression: \( 4 - 5 \sqrt{10} + 7 \).

Step 3: Perform the Addition and Subtraction

Add and subtract the numbers in the expression. Simplify it stepwise:1. Add 4 and 7 to get 11.2. The expression is now: \( 11 - 5 \sqrt{10} \).

Step 4: Combine Terms

Since no like terms remain (as \( 5 \sqrt{10} \) is a different term from the constant), the expression simplifies to: \( 11 - 5 \sqrt{10} \).

Key Concepts

Square Roots

Square roots are quite fascinating! They represent a number that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 multiplied by 4 equals 16.
Square roots are usually expressed with a radical symbol (√). When you see \(\sqrt{16}\), you know you're finding what number multiplied by itself is equal to 16.
If the number under the radical sign is a perfect square, meaning it can be expressed as the square of a whole number, the square root will be a whole number too. This is important in simplifying expressions.

Perfect Squares

Perfect squares are numbers that have integer square roots. In simpler terms, when you take the square root of a perfect square, the result is a whole number.
Examples of perfect squares include 1, 4, 9, 16, etc. Notice how these numbers come from squaring whole numbers: \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\), and so on.
Identifying perfect squares in expressions, like \(\sqrt{16}\), helps simplify expressions quickly, as you substitute the number with its integer root.

Addition and Subtraction of Radicals

When dealing with addition or subtraction with radicals, it’s similar to combining like terms in algebra; only like radicals can be combined. This means the radicals must be the same in both number and radicand (the number under the radical sign).
For instance, you can add \(3\sqrt{5}\) and \(2\sqrt{5}\) to get \(5\sqrt{5}\), but you can't directly add \(3\sqrt{5}\) and \(2\sqrt{3}\), because the radicands aren't the same.
In our given expression, unfortunately, \(5\sqrt{10}\) will just remain as it is since it doesn't match with any other radical. The constants will be the only numbers you can simply add or subtract, which is a neat trick to remember!

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. Understanding how to work with these is a fundamental skill in algebra.
Expressions can include constants (like 3 or 7) and coefficients with variables and radicals. Simplifying them involves performing operations like addition, subtraction, multiplication, division, and sometimes factoring.
In the expression \(11 - 5\sqrt{10}\), we have constants (11) and a term with a radical and a coefficient (-5\(\sqrt{10}\)). Knowing when and how to combine these elements is key in algebra.