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Interactive multiple-choice quiz: Multiplying & Dividing Radical Expressions - Reveal
Simplifying expressions involving radicals is a fundamental algebraic skill. This process includes using the product and quotient rules, rationalizing denominators to remove radicals from the bottom of a fraction, and applying the distributive property to multiply binomials containing roots. Mastery of these techniques ensures that mathematical expressions are presented in their most standard and concise form.
🏆 انضم إلى التحدي واحصل على ترتيبك
اختبار شهادة تدريبي مؤقت للصف والمادة والفصل نفسه.
ابدأ السباق ✨
اختر إجابة واحدة لكل سؤال. عند الاختيار ستظهر النتيجة فورًا: الأخضر صحيح، والأحمر خطأ، وسيظهر تفسير الإجابة مباشرة إن كان متوفرًا. وبعد آخر سؤال ستظهر الدرجة النهائية تلقائيًا.
🚩 Report
Multiply and simplify. Assume that all variables are positive. \(\sqrt{8y^5} \cdot \sqrt{40y^2}\)
A
\(8y^3 \sqrt{5}\)
B
\(8y^3 \sqrt{5y}\)
C
\(8y \sqrt{5y}\)
D
\(8y^3 \sqrt{5}\)
Explanation
Multiply the radicands: 8y5 × 40y2 = 320y7 . Then simplify: \(\sqrt{320y^7} = \sqrt{64 \cdot 5 \cdot y^6 \cdot y} = 8y^3 \sqrt{5y}\).
🚩 Report
Divide and simplify. Assume that all variables are positive. \(\frac{\sqrt{600}}{\sqrt{6}}\)
Explanation
Use the quotient property of radicals: \(\frac{\sqrt{600}}{\sqrt{6}} = \sqrt{\frac{600}{6}} = \sqrt{100} = 10\).
🚩 Report
Divide and simplify. Assume that all variables are positive. \(\frac{\sqrt{180x^5}}{\sqrt{5x^3}}\)
Explanation
Divide the terms under the radical: \(\sqrt{\frac{180x^5}{5x^3}} = \sqrt{36x^2} = 6x\).
🚩 Report
Simplify \((\sqrt{2} - 3)(\sqrt{2} + 3)\)
A
7
B
-7
C
\(\sqrt{7}\)
D
\(-\sqrt{7}\)
Explanation
Apply the difference of squares formula (a-b)(a+b) = a2 - b2 . Thus, \((\sqrt{2})^2 - 3^2 = 2 - 9 = -7\).
🚩 Report
Simplify \((\sqrt{5} - 7)^2\)
A
\(54 - \sqrt{70}\)
B
35
C
\(54 - 14\sqrt{5}\)
D
\(\sqrt{35}\)
Explanation
Apply the identity (a-b)2 = a2 - 2ab + b2 . This gives \((\sqrt{5})^2 - 2(7)(\sqrt{5}) + 7^2 = 5 - 14\sqrt{5} + 49 = 54 - 14\sqrt{5}\).
🚩 Report
Simplify \(\frac{1}{2 + \sqrt{3}}\)
A
1
B
\(2 - \sqrt{3}\)
C
\(\frac{\sqrt{3}}{2}\)
D
\(\frac{1}{\sqrt{3}}\)
Explanation
Multiply the numerator and denominator by the conjugate \(2 - \sqrt{3}\) to rationalize: \(\frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}\).
🚩 Report
Simplify \(\frac{2 + \sqrt{6}}{3 - \sqrt{6}}\)
A
\(\frac{12 + 5\sqrt{6}}{3}\)
B
\(\frac{2}{3}\)
C
\(\frac{3}{2}\)
D
undefined
Explanation
Multiply numerator and denominator by the conjugate \(3 + \sqrt{6}\): \(\frac{(2 + \sqrt{6})(3 + \sqrt{6})}{(3 - \sqrt{6})(3 + \sqrt{6})} = \frac{6 + 2\sqrt{6} + 3\sqrt{6} + 6}{9 - 6} = \frac{12 + 5\sqrt{6}}{3}\).
🚩 Report
Simplify \(\frac{1}{\sqrt{2}}\)
A
1
B
2
C
\(\sqrt{2}\)
D
\(\frac{\sqrt{2}}{2}\)
Explanation
Rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{2}\): \(\frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2}\).
🚩 Report
Simplify \(\sqrt{2}(5 + \sqrt{2})\)
A
9
B
\(5\sqrt{2} + 2\)
C
\(10\sqrt{2}\)
D
\(2\sqrt{2} + 3\sqrt{30}\)
Explanation
Distribute \(\sqrt{2}\) into the parentheses: \(\sqrt{2} \cdot 5 + \sqrt{2} \cdot \sqrt{2} = 5\sqrt{2} + 2\).
🚩 Report
\(\sqrt{5} \times \sqrt{10}\)
A
50
B
\(2\sqrt{5}\)
C
\(5\sqrt{2}\)
D
\(5\sqrt{10}\)
Explanation
\(\sqrt{5} \cdot \sqrt{10} = \sqrt{50}\). Simplifying gives \(\sqrt{25 \cdot 2} = 5\sqrt{2}\).
🚩 Report
Simplify: \(6\sqrt{2} \cdot 5\sqrt{14}\)
A
\(32\sqrt{7}\)
B
\(60\sqrt{7}\)
C
\(2\sqrt{7}\)
D
\(30\sqrt{7}\)
Explanation
Multiply coefficients and radicands: \((6 \cdot 5)\sqrt{2 \cdot 14} = 30\sqrt{28}\). Simplify \(30\sqrt{4 \cdot 7} = 30 \cdot 2\sqrt{7} = 60\sqrt{7}\).
🚩 Report
Simplify: \(3\sqrt{5}(-5\sqrt{5} + 4)\)
A
\(4\sqrt{3} + 3\)
B
\(6\sqrt{3}\)
C
\(-75 + 12\sqrt{5}\)
D
\(-10\sqrt{5}\)
Explanation
Distribute \(3\sqrt{5}\): \(3\sqrt{5} \cdot (-5\sqrt{5}) + 3\sqrt{5} \cdot 4 = -15(5) + 12\sqrt{5} = -75 + 12\sqrt{5}\).
🚩 Report
Simplify \((2 + \sqrt{2})(-1 + \sqrt{2})\)
A
\(6 + 5\sqrt{5}\)
B
\(4\sqrt{3} + 2\)
C
\(15 + \sqrt{10} + 3\sqrt{15} + \sqrt{6}\)
D
\(\sqrt{2}\)
Explanation
Expand the binomials: \(2(-1) + 2\sqrt{2} - \sqrt{2} + (\sqrt{2})^2 = -2 + \sqrt{2} + 2 = \sqrt{2}\).
🚩 Report
Simplify \(\frac{3\sqrt{20}}{\sqrt{45}}\)
A
2
B
\(\frac{1}{2}\)
C
\(\frac{1}{3}\)
D
\(\sqrt{2}\)
Explanation
Simplify the radicals first: \(\frac{3 \cdot 2\sqrt{5}}{3\sqrt{5}} = \frac{6\sqrt{5}}{3\sqrt{5}} = 2\).
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