Problem 100
Question
Simplify. $$ \frac{15+\sqrt{75}}{5} $$
Step-by-Step Solution
Verified Answer
3 + \( \sqrt{3} \)
1Step 1: Simplify the square root
First, simplify the square root expression. Notice that \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\). So the given expression becomes \(\frac{15 + 5\sqrt{3}}{5}\).
2Step 2: Split the fraction
Next, split the fraction into two separate fractions: \( \frac{15}{5} + \frac{5\sqrt{3}}{5}\).
3Step 3: Simplify each fraction
Simplify each fraction separately. \(\frac{15}{5} = 3\) and \( \frac{5\sqrt{3}}{5} = \sqrt{3}\). So, the expression simplifies to \(3 + \sqrt{3}\).
Key Concepts
square rootsfractionssimplification steps
square roots
Square roots are an essential part of algebra. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, \(\text{the square root of 25 is 5}\), because \(5 \times 5 = 25\).
In our example, we are given \(\sqrt{75}\). We can simplify \(\sqrt{75}\) by breaking it down into its prime factors. Notice that \(75 = 25 \times 3\). Since \(\sqrt{25} = 5\), we can simplify \(\sqrt{75}\) to \(5\sqrt{3}\).
So, simplifying \(\sqrt{75}\) is our first step:
\(\sqrt{75} = 5\sqrt{3}\).
This simplification makes it easier to work with square roots when simplifying algebraic expressions.
In our example, we are given \(\sqrt{75}\). We can simplify \(\sqrt{75}\) by breaking it down into its prime factors. Notice that \(75 = 25 \times 3\). Since \(\sqrt{25} = 5\), we can simplify \(\sqrt{75}\) to \(5\sqrt{3}\).
So, simplifying \(\sqrt{75}\) is our first step:
\(\sqrt{75} = 5\sqrt{3}\).
This simplification makes it easier to work with square roots when simplifying algebraic expressions.
fractions
Fractions represent parts of a whole and are often seen in algebra. Simplifying fractions is crucial to solving algebraic expressions.
A fraction consists of a numerator (top part) and a denominator (bottom part). In our example, we start with \(\frac{15 + \sqrt{75}}{5}\).
After simplifying the square root, this turns into \(\frac{15 + 5\sqrt{3}}{5}\).
Next, we split the fraction into two parts: \(\frac{15}{5} + \frac{5\sqrt{3}}{5}\). This is possible because each term in the numerator can be divided by the denominator separately.
By splitting the fraction, we simplify each part individually.
This step-by-step approach ensures a clear and correct final expression.
A fraction consists of a numerator (top part) and a denominator (bottom part). In our example, we start with \(\frac{15 + \sqrt{75}}{5}\).
After simplifying the square root, this turns into \(\frac{15 + 5\sqrt{3}}{5}\).
Next, we split the fraction into two parts: \(\frac{15}{5} + \frac{5\sqrt{3}}{5}\). This is possible because each term in the numerator can be divided by the denominator separately.
By splitting the fraction, we simplify each part individually.
This step-by-step approach ensures a clear and correct final expression.
simplification steps
Simplification is an important process in algebra to make expressions more manageable and easier to understand.
Our given expression can be broken down into several simplification steps:
Finally, combine the simplified parts to get the final expression: \(\text{3 + \sqrt{3}}\).
These steps ensure clarity and accuracy, making complex expressions much simpler to handle.
Our given expression can be broken down into several simplification steps:
- Step 1: **Simplify the square root.** Convert \(\sqrt{75}\) into \(5\sqrt{3}\).
- Step 2: **Split the fraction.** Rewrite \(\frac{15 + 5\sqrt{3}}{5}\) as \(\frac{15}{5} + \frac{5\sqrt{3}}{5}\).
- Step 3: **Simplify each fraction.** Calculate each part separately: \(\frac{15}{5}=3\) and \(\frac{5\sqrt{3}}{5}=\sqrt{3}\).
Finally, combine the simplified parts to get the final expression: \(\text{3 + \sqrt{3}}\).
These steps ensure clarity and accuracy, making complex expressions much simpler to handle.