Simplify each radical by removing as many factors as possible: Square root of 75, square root of 72, square root of 162
Understand the Problem
The question asks to simplify three radical expressions (square roots) by factoring out perfect squares from the radicands (the numbers inside the square roots) and taking their square roots.
Answer
$\sqrt{8} = 2\sqrt{2}$ $\sqrt{18} = 3\sqrt{2}$ $\sqrt{32} = 4\sqrt{2}$
Answer for screen readers
$\sqrt{8} = 2\sqrt{2}$
$\sqrt{18} = 3\sqrt{2}$
$\sqrt{32} = 4\sqrt{2}$
Steps to Solve
- Simplify $\sqrt{8}$
Factor 8 into its prime factors: $8 = 2 \cdot 2 \cdot 2 = 2^2 \cdot 2$. Then, $\sqrt{8} = \sqrt{2^2 \cdot 2} = \sqrt{2^2} \cdot \sqrt{2} = 2\sqrt{2}$.
- Simplify $\sqrt{18}$
Factor 18 into its prime factors: $18 = 2 \cdot 3 \cdot 3 = 2 \cdot 3^2$. Then, $\sqrt{18} = \sqrt{2 \cdot 3^2} = \sqrt{3^2} \cdot \sqrt{2} = 3\sqrt{2}$.
- Simplify $\sqrt{32}$
Factor 32 into its prime factors: $32 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^5 = 2^4 \cdot 2 = (2^2)^2 \cdot 2 = 4^2 \cdot 2$. Then, $\sqrt{32} = \sqrt{4^2 \cdot 2} = \sqrt{4^2} \cdot \sqrt{2} = 4\sqrt{2}$.
$\sqrt{8} = 2\sqrt{2}$
$\sqrt{18} = 3\sqrt{2}$
$\sqrt{32} = 4\sqrt{2}$
More Information
Simplifying radicals by factoring out perfect squares is a useful technique in algebra and calculus. It allows us to express radicals in their simplest form, making it easier to perform operations such as addition, subtraction, multiplication, and division.
Tips
A common mistake is not fully factoring the number inside the square root, which can lead to not extracting the largest possible perfect square. For example, with $\sqrt{32}$, one might initially factor it as $\sqrt{4 \cdot 8}$ and simplify to $2\sqrt{8}$. However, $\sqrt{8}$ can be further simplified to $2\sqrt{2}$, so the complete simplification is $2 \cdot 2\sqrt{2} = 4\sqrt{2}$. To avoid this, always ensure you have factored out the largest perfect square.
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