Question
Simplify the following expression. \( \sqrt{5} \cdot \sqrt{50} \)
Ask by Gray Zimmerman. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(5\sqrt{10}\)
Solution
To simplify the expression \( \sqrt{5} \cdot \sqrt{50} \), follow these steps:
1. **Use the Product Property of Square Roots:**
The product of two square roots can be combined under a single square root:
\[
\sqrt{5} \cdot \sqrt{50} = \sqrt{5 \times 50} = \sqrt{250}
\]
2. **Simplify \( \sqrt{250} \):**
Factor 250 into its prime factors to identify perfect squares:
\[
250 = 25 \times 10 = 5^2 \times 10
\]
Now, apply the square root to each factor:
\[
\sqrt{250} = \sqrt{5^2 \times 10} = \sqrt{5^2} \times \sqrt{10} = 5\sqrt{10}
\]
So, the simplified form of \( \sqrt{5} \cdot \sqrt{50} \) is:
\[
5\sqrt{10}
\]
**Answer:** \(5\sqrt{10}\)
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Simplify this solution The Deep Dive
To simplify \( \sqrt{5} \cdot \sqrt{50} \), you can use the property of square roots that states \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). So, first, we multiply inside the square root: \[ \sqrt{5} \cdot \sqrt{50} = \sqrt{5 \cdot 50} = \sqrt{250} \] Next, we can simplify \( \sqrt{250} \). We can factor 250 into prime factors: \[ 250 = 25 \cdot 10 = 5^2 \cdot 10 \] This means: \[ \sqrt{250} = \sqrt{5^2 \cdot 10} = \sqrt{5^2} \cdot \sqrt{10} = 5\sqrt{10} \] So, the simplified expression is: \[ 5\sqrt{10} \]
