Practice Problems

Problems:

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Problem 1:

Explanation:

Break down $50$ into factors where one of the factors is a perfect square. Then, simplify the square root of the perfect square.

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Problem 2:

Explanation:

Simplify each surd separately by breaking down $18$ and $8$ into factors. Then, add the simplified surds if possible.

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Problem 3:

Explanation:

Multiply the numbers inside the square roots first, then simplify the result if it's a perfect square.

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Problem 4:

Explanation:

Divide the numbers inside the square roots, then simplify the resulting square root.

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Problem 5:

Explanation:

Simplify each surd individually, then add or subtract the surds as needed. Combine like terms if possible.

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Solutions:

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Problem 1:

• Step 1: Break down $50$ into factors where one of them is a perfect square.
  • $50 = 25 \times 2$
  • Explanation: We choose $25$ because it is a perfect square (its square root is a whole number), which will help simplify the surd.
• Step 2: Take the square root of the perfect square.
  • $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2}$
  • Explanation: We split the square root into two parts: the square root of $25$ and the square root of $2$. This makes it easier to simplify.
• Step 3: Simplify the square root of the perfect square.
  • $\sqrt{25} = 5$, so $\sqrt{50} = 5\sqrt{2}$
  • Explanation: We know that the square root of $25$ is $5$, so we replace $\sqrt{25}$ with $5$. The $\sqrt{2}$ stays as it is because $2$ is not a perfect square.
• Final Answer: $\sqrt{50} = :success[5\sqrt{2}]$

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Problem 2:

• Step 1: Simplify each surd separately.
  • For $\sqrt{18}$:
  • $\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}$
  • Explanation: We break down $18$ into $9$ and $2$ because 9 is a perfect square. We then simplify $\sqrt{9}$ to $3$ and keep $\sqrt{2}$ as it is.
  • For $\sqrt{8}$:
  • $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$
  • Explanation: Similarly, we break down $8$ into $4$ and $2$, simplify $\sqrt{4}$ to $2$, and keep $\sqrt{2}$ as it is.
• Step 2: Add the simplified surds.
  • $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$
  • Explanation: Since both terms have $\sqrt{2}$, we can add the numbers in front of the surds ($3$ and $2$ ) to get $5$.
• Final Answer:$\sqrt{18} + \sqrt{8} = :success[5\sqrt{2}]$

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Problem 3:

• Step 1: Multiply the numbers inside the square roots.
  • $\sqrt{3} \times \sqrt{12} = \sqrt{3 \times 12} = \sqrt{36}$
  • Explanation: We multiply $3$ and $12$ inside the square roots to get $36$. Combining the square roots into one allows us to simplify more easily.
• Step 2: Simplify the square root of the product.
  • $\sqrt{36} = 6$
  • Explanation: Since $36$ is a perfect square, we can take its square root, which is $6$.
• Final Answer: $\sqrt{3} \times \sqrt{12} = :success[6]$

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Problem 4:

• Step 1: Divide the numbers inside the square roots.
  • $\frac{\sqrt{75}}{\sqrt{3}} = \sqrt{\frac{75}{3}} = \sqrt{25}$
  • Explanation: We divide $75$ by $3$ inside the square root to get $25$, which is a perfect square. This division step simplifies the expression.
• Step 2: Simplify the square root of the quotient.
  • $\sqrt{25} = 5$
  • Explanation: Since $25$ is a perfect square, we take its square root, which is $5$.
• Final Answer: $\frac{\sqrt{75}}{\sqrt{3}} = :success[5]$

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Problem 5:

• Step 1: Simplify each surd separately.
  • $\sqrt{50} = 5\sqrt{2}$ (as in Problem $1$)
  • $\sqrt{18} = 3\sqrt{2}$ (as in Problem $2$)
  • $\sqrt{8} = 2\sqrt{2}$ (as in Problem $2$)
• Step 2: Combine the simplified surds by adding and subtracting.
  • $5\sqrt{2} + 3\sqrt{2} - 2\sqrt{2} = (5 + 3 - 2)\sqrt{2} = 6\sqrt{2}$
  • Explanation: First, add $5$ and $3$ to get $8$, then subtract $2$ to get $6$. All the terms involve $\sqrt{2}$, so we combine the numbers in front.
• Final Answer: $\sqrt{50} + \sqrt{18} - \sqrt{8} = :success[6\sqrt{2}]$

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