Determine if $$ \sqrt{75} $$ is rational or irrational and give a reason for your answe

Question

UpStudy AI · Accepted Answer

$$ \sqrt{75} $$ is **irrational**. Here's why:

1. **Simplifying the Square Root:**
   $$
   \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}
   $$
   
2. **Understanding Irrational Numbers:**
   - A number is **rational** if it can be expressed as a fraction $$ \frac{a}{b} $$, where $$ a $$ and $$ b $$ are integers and $$ b \neq 0 $$.
   - A number is **irrational** if it cannot be expressed in such a way.

3. **Analyzing $$ \sqrt{3} $$:**
   - It's a well-known fact in mathematics that $$ \sqrt{3} $$ is irrational. There are several proofs for this, one of which involves assuming the opposite (that $$ \sqrt{3} $$ is rational) and reaching a contradiction.

4. **Conclusion for $$ \sqrt{75} $$:**
   - Since $$ \sqrt{75} = 5\sqrt{3} $$ and $$ \sqrt{3} $$ is irrational, multiplying an irrational number by a non-zero rational number (in this case, 5) results in an irrational number.
   - Therefore, $$ \sqrt{75} $$ cannot be expressed as a simple fraction of two integers, making it **irrational**.

**Summary:** $$ \sqrt{75} $$ is irrational because it simplifies to $$ 5\sqrt{3} $$, and $$ \sqrt{3} $$ is an irrational number.
$$ \sqrt{75} $$ is irrational because it simplifies to $$ 5\sqrt{3} $$, and $$ \sqrt{3} $$ is an irrational number.

Determine if \( \sqrt{75} \) is rational or irrational and give a reason for your answe

Real Tutor Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Pre Algebra Questions