Add or subtract, as indicated. See Example 6. √45 + 4√20

Verified step by step guidance

1. Step 1: Simplify each square root separately. Start with $\sqrt{45}$.
2. Step 2: Factor 45 into its prime factors: $45=9\times 5$.
3. Step 3: Recognize that 9 is a perfect square, so $\sqrt{45}=\sqrt{9\times 5}=\sqrt{9}\times \sqrt{5}=3\sqrt{5}$.
4. Step 4: Simplify $4\sqrt{20}$ by factoring 20 into its prime factors: $20=4\times 5$.
5. Step 5: Recognize that 4 is a perfect square, so $4\sqrt{20}=4\times \sqrt{4\times 5}=4\times 2\sqrt{5}=8\sqrt{5}$.

Key Concepts

Simplifying Square Roots

Simplifying square roots involves breaking down a square root into its prime factors to express it in a simpler form. For example, √45 can be simplified to √(9 × 5) = 3√5, as 9 is a perfect square. This process is essential for combining like terms in expressions involving square roots.

Combining Like Terms

Combining like terms is a fundamental algebraic skill that involves adding or subtracting terms that have the same variable or radical part. In the expression √45 + 4√20, after simplifying the square roots, you can combine the resulting terms if they share the same radical component, which streamlines the expression.

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing for the multiplication of a single term across a sum or difference. In the context of square roots, this property can be applied when dealing with coefficients in front of radicals, such as 4√20, to facilitate simplification and combination of terms.