Simplify $\sqrt{5}(\sqrt{15}-\sqrt{3})$
Understand the Problem
The question requires simplifying an expression involving the product of a square root with a difference of square roots. We need to apply the distributive property and simplify each term by factoring out perfect squares from under the square root.
Answer
-2
Answer for screen readers
-2
Steps to Solve
- Apply the distributive property
Multiply $\sqrt{2}$ by each term inside the parentheses:
$$ \sqrt{2}(\sqrt{8} - \sqrt{18}) = \sqrt{2} \cdot \sqrt{8} - \sqrt{2} \cdot \sqrt{18} $$
- Simplify the products of square roots
Using the property $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, simplify each term:
$$ \sqrt{2 \cdot 8} - \sqrt{2 \cdot 18} = \sqrt{16} - \sqrt{36} $$
- Evaluate the square roots
Calculate the square roots of 16 and 36:
$$ \sqrt{16} = 4 $$ $$ \sqrt{36} = 6 $$
So the expression becomes:
$$ 4 - 6 $$
- Perform the subtraction
Subtract 6 from 4:
$$ 4 - 6 = -2 $$
-2
More Information
The simplified expression is -2. This is an integer.
Tips
A common mistake is not simplifying the square roots correctly. For example, not recognizing that $\sqrt{16} = 4$ and $\sqrt{36} = 6$. Another mistake could be an arithmetic error when subtracting.
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