Problem 18 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \sqrt{16}-5 \sqrt{10}+7 $$

Step-by-Step Solution

Answer

The solution is \( 11 - 5 \sqrt{10} \).

Step 1: Simplify the square root

First, let's simplify the square root expression \( \sqrt{16} \). Because 16 is a perfect square, we can simplify it to 4.

Step 2: Substitute simplified value

Now that we know \( \sqrt{16} = 4 \), we can substitute this back into the original expression. Thus, the expression becomes \( 4 - 5 \sqrt{10} + 7 \).

Step 3: Combine like terms

The expression \( 4 - 5 \sqrt{10} + 7 \) consists of two constant terms \( 4 \) and \( 7 \). Add these constants together: \( 4 + 7 = 11 \). So the expression simplifies to \( 11 - 5 \sqrt{10} \).

Step 4: Write the final answer

The simplified expression of the original problem is \( 11 - 5 \sqrt{10} \). Therefore, this is our final answer.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, especially when working with radicals. A square root asks "What number multiplied by itself gives me this number?" For example, the square root of 16, which we write as \( \sqrt{16} \), asks us to find a number that, when squared, equals 16. The answer is 4, because \( 4 \times 4 = 16 \). Similarly, if we work with square roots beyond perfect squares, they might not simplify into neat whole numbers.Square roots can often be simplified by identifying perfect square factors. For example, if you handle \( \sqrt{18} \), you can break it down into \( \sqrt{9 \times 2} \), which simplifies to \( 3\sqrt{2} \) because 9 is a perfect square, and its square root is 3.Understanding square roots is crucial in algebra, as it allows us to rewrite and simplify expressions with radicals.

Combining Like Terms

Combining like terms is a process that simplifies expressions and equations in algebra. Like terms are terms that have identical variable parts, ensuring they can be added or subtracted. However, it's essential to notice if coefficients are numbers, plain, or if they're attached to square roots.For instance, in the expression \( 4 - 5\sqrt{10} + 7 \), the numbers 4 and 7 are like terms because they are both constants—plain numbers without a variable attached. We can combine them to simplify the expression to \( 11 - 5\sqrt{10} \).Keep in mind:

• Only like terms can be combined.
• Terms like \( 2x \) and \( 3y \) cannot be combined unless they have the same variable part.
• If a term has a square root, you can only combine it with other terms that have the same square root.

Perfect Squares

A perfect square is a number that results from a whole number multiplied by itself. For example, the number 16 is a perfect square because it equals \( 4 \times 4 \). Perfect squares are crucial in simplifying square roots because they allow square roots to resolve into whole numbers easily.Here are a few common perfect squares you should recognize:

• \(1\), \(4\), \(9\), \(16\), \(25\), \(36\)
• \(49\), \(64\), \(81\), \(100\), and many more

When you encounter a square root in an expression, checking if the number is a perfect square will guide you in simplifying it effectively. Recognizing perfect squares allows you to quickly pull out a square root and simplify a mathematical expression, streamlining your computations in algebra and beyond.