Basic to Moderate (1-20)
▶1. What is a surd? Give one example.
Answer: A surd is an irrational root that cannot be simplified to a rational number.
Explanation:
- For example, \( \sqrt{2} \) is a surd because it cannot be written as a simple fraction and its decimal goes on forever without repeating.
- But \( \sqrt{4} = 2 \) is not a surd, because it is a whole number.
▶2. Is \( \sqrt{9} \) a surd? Why or why not?
Answer: No, \( \sqrt{9} = 3 \) is not a surd.
Explanation:
- \( \sqrt{9} = 3 \), which is a rational number (a whole number).
- Surds are only for roots that are irrational (cannot be written as a fraction or whole number).
▶3. Simplify: \( 2\sqrt{3} + 5\sqrt{3} \)
Answer: \( 7\sqrt{3} \)
Explanation:
- Both terms have the same surd (\( \sqrt{3} \)), so you can add the numbers in front.
- \( 2\sqrt{3} + 5\sqrt{3} = (2+5)\sqrt{3} = 7\sqrt{3} \)
▶4. Simplify: \( 4\sqrt{2} - 3\sqrt{2} \)
Answer: \( \sqrt{2} \)
Explanation:
- Both terms have \( \sqrt{2} \), so subtract the numbers in front.
- \( 4\sqrt{2} - 3\sqrt{2} = (4-3)\sqrt{2} = 1\sqrt{2} = \sqrt{2} \)
▶5. Simplify: \( \sqrt{18} \)
Answer: \( 3\sqrt{2} \)
Explanation:
- Break 18 into 9 × 2: \( \sqrt{18} = \sqrt{9 \times 2} \)
- \( \sqrt{9} = 3 \), so \( \sqrt{18} = 3\sqrt{2} \)
▶6. Simplify: \( \sqrt{50} \)
Answer: \( 5\sqrt{2} \)
Explanation:
- Break 50 into 25 × 2: \( \sqrt{50} = \sqrt{25 \times 2} \)
- \( \sqrt{25} = 5 \), so \( \sqrt{50} = 5\sqrt{2} \)
▶7. Simplify: \( 2\sqrt{5} \times 3\sqrt{2} \)
Answer: \( 6\sqrt{10} \)
Explanation:
- Multiply the numbers in front: 2 × 3 = 6
- Multiply the surds: \( \sqrt{5} \times \sqrt{2} = \sqrt{10} \)
- So, \( 2\sqrt{5} \times 3\sqrt{2} = 6\sqrt{10} \)
▶8. Simplify: \( \sqrt{12} + \sqrt{27} \)
Answer: \( 5\sqrt{3} \)
Explanation:
- \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \)
- \( \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \)
- So, \( 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3} \)
▶9. Simplify: \( \sqrt{8} - \sqrt{2} \)
Answer: \( \sqrt{8} = 2\sqrt{2} \), so \( 2\sqrt{2} - \sqrt{2} = \sqrt{2} \)
Explanation:
- \( \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \)
- \( 2\sqrt{2} - \sqrt{2} = (2-1)\sqrt{2} = \sqrt{2} \)
▶10. Simplify: \( \sqrt{45} + \sqrt{20} \)
Answer: \( 3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5} \)
Explanation:
- \( \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \)
- \( \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \)
- So, \( 3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5} \)
▶11. Multiply: \( \sqrt{3} \times \sqrt{12} \)
Answer: \( 6 \)
Explanation:
- \( \sqrt{3} \times \sqrt{12} = \sqrt{3 \times 12} = \sqrt{36} = 6 \)
▶12. Divide: \( 6\sqrt{8} \div 2\sqrt{2} \)
Answer: \( 6\sqrt{8} \div 2\sqrt{2} = 3\sqrt{4} = 6 \)
Explanation:
- Divide the numbers: 6 ÷ 2 = 3
- Divide the surds: \( \sqrt{8} \div \sqrt{2} = \sqrt{8/2} = \sqrt{4} = 2 \)
- So, \( 3 \times 2 = 6 \)
▶13. Rationalize the denominator: \( \frac{3}{\sqrt{5}} \)
Answer: \( \frac{3\sqrt{5}}{5} \)
Explanation:
- Multiply top and bottom by \( \sqrt{5} \):
- \( \frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5} \)
▶14. Rationalize the denominator: \( \frac{1}{\sqrt{2} + 1} \)
Answer: \( \sqrt{2} - 1 \)
Explanation:
- Multiply top and bottom by the conjugate (\( \sqrt{2} - 1 \)):
- \( \frac{1}{\sqrt{2} + 1} \times \frac{\sqrt{2} - 1}{\sqrt{2} - 1} = \frac{\sqrt{2} - 1}{(\sqrt{2})^2 - 1^2} = \frac{\sqrt{2} - 1}{2 - 1} = \sqrt{2} - 1 \)
▶15. What is the conjugate of \( 2 + 3\sqrt{5} \)?
Answer: \( 2 - 3\sqrt{5} \)
Explanation:
- The conjugate is made by changing the plus to minus: \( 2 + 3\sqrt{5} \rightarrow 2 - 3\sqrt{5} \)
▶16. Simplify: \( (\sqrt{3} + 2)^2 \)
Answer: \( 7 + 4\sqrt{3} \)
Explanation:
- Use the formula \( (a+b)^2 = a^2 + 2ab + b^2 \)
- \( (\sqrt{3})^2 = 3 \), \( 2^2 = 4 \), \( 2 \times \sqrt{3} \times 2 = 4\sqrt{3} \)
- So, \( 3 + 4 + 4\sqrt{3} = 7 + 4\sqrt{3} \)
▶17. Simplify: \( (\sqrt{5} - 1)^2 \)
Answer: \( 6 - 2\sqrt{5} \)
Explanation:
- Use the formula \( (a-b)^2 = a^2 - 2ab + b^2 \)
- \( (\sqrt{5})^2 = 5 \), \( 1^2 = 1 \), \( 2 \times \sqrt{5} \times 1 = 2\sqrt{5} \)
- So, \( 5 - 2\sqrt{5} + 1 = 6 - 2\sqrt{5} \)
▶18. If \( \sqrt{a} = 5 \), what is the value of \( a \)?
Answer: \( a = 25 \)
Explanation:
- Square both sides: \( (\sqrt{a})^2 = 5^2 \)
- \( a = 25 \)
▶19. If \( \sqrt{49x} = 21 \), what is the value of \( x \)?
Answer: \( x = 9 \)
Explanation:
- Square both sides: \( 49x = 21^2 = 441 \)
- \( x = 441 \div 49 = 9 \)
▶20. True or False: Every square root is a surd.
Answer: False.
Explanation:
- Only square roots that are irrational (like \( \sqrt{2} \), \( \sqrt{3} \)) are surds.
- Square roots that are whole numbers (like \( \sqrt{4} = 2 \)) are not surds.