Problem 36 - Intermediate Algebra

Question

Add or subtract. $$ \frac{\sqrt{45}}{10}+\frac{7 \sqrt{5}}{10} $$

Step-by-Step Solution

Answer

The result is \( \sqrt{5} \).

Step 1: Simplify Square Roots

The expression contains \( \sqrt{45} \). Notice that \( 45 = 9 \times 5 \), so \( \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \). Thus, the expression becomes \( \frac{3\sqrt{5}}{10} + \frac{7\sqrt{5}}{10} \).

Step 2: Combine Like Terms

Since both terms have the same denominator and the same 'like' terms (\( \sqrt{5} \)), you can add them directly. Combine the numerators: \( 3\sqrt{5} + 7\sqrt{5} = 10\sqrt{5} \).

Step 3: Simplify the Result

Substitute the combined numerators back into the expression: \( \frac{10\sqrt{5}}{10} \). Now, since both the numerator and the denominator are divisible by 10, the expression simplifies to \( \sqrt{5} \).

Key Concepts

Simplifying Square Roots

When you encounter a square root, like \( \sqrt{45} \), the first step toward simplifying it is to factor the number inside the square root into its prime components. For \( 45 \), the prime factorization is \( 9 \times 5 \). You should notice that 9 is a perfect square (since \( 9 = 3 \times 3 \)).
Therefore, \( \sqrt{45} \) can be rewritten as \( \sqrt{9 \times 5} \), which simplifies further to \( \sqrt{9} \times \sqrt{5} \).
This results in \( 3\sqrt{5} \) because \( \sqrt{9} = 3 \).

Simplifying square roots helps in further calculations by reducing complex square root terms to more manageable numbers. This is especially useful when dealing with expressions involving multiple radicals, as simplified terms are simpler to work with in arithmetic operations.

Like Terms

In mathematics, 'like terms' refer to terms whose variables and their exponents are exactly the same. When dealing with radicals, terms such as \( 3\sqrt{5} \) and \( 7\sqrt{5} \) can be considered 'like' because they both contain the radical \( \sqrt{5} \). This common feature allows us to combine them through addition or subtraction, just as if they were simple algebraic terms.

It’s important to identify like terms because only these can be added or subtracted directly. When we look at the expression \( \frac{3\sqrt{5}}{10} + \frac{7\sqrt{5}}{10} \), we recognize that both terms are like terms due to the common radical and denominator. Therefore, combining them is as simple as adding normal numbers—since both have \( \sqrt{5} \), you simply add their coefficients: \( 3 + 7 \).
By understanding this concept, you can streamline the solving process for more complex expressions by focusing only on combining terms that share identical radicals.

Combining Radicals

Combining radicals involves both understanding the nature of like terms and applying arithmetic to these terms. When you have radicals with the same index and radicand, you can add or subtract them as you would with other algebraic terms.
In our exercise, after simplifying \( \sqrt{45} \) to \( 3\sqrt{5} \), we have an expression with two fractions: \( \frac{3\sqrt{5}}{10} \) and \( \frac{7\sqrt{5}}{10} \).
Since they have both the same radical term \( \sqrt{5} \) and the same denominator, they are ripe for combining.
Here’s how to combine them:

• Make sure both radicals are simplified, as seen previously.
• Ensure all terms are like terms—both numerators here have \( \sqrt{5} \).
• Add the coefficients of the radical terms in the numerators: \( 3 + 7 = 10 \).

Thus, the expression \( \frac{3\sqrt{5}}{10} + \frac{7\sqrt{5}}{10} \) becomes \( \frac{10\sqrt{5}}{10} \).
Lastly, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, reducing the expression to just \( \sqrt{5} \). By mastering combining radicals, you simplify expressions more efficiently, making advanced problems much more approachable.