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Interactive multiple-choice quiz: revision on nth roots and rational exponent - Reveal
This worksheet provides a comprehensive review of nth roots and rational exponents. It covers converting between radical and exponential forms, simplifying complex algebraic expressions with fractional exponents, and solving radical equations by isolating the radical. Key concepts include identifying real roots for various indices and applying exponent laws to simplify products and quotients of terms with rational powers.
🏆 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
Rewrite into rational exponent form: \( \sqrt{10} \)
Explanation
A square root can be written as a rational exponent with an index of 2, so \( \sqrt{10} = 10^{1/2} \).
🚩 Report
Simplify:
323/5
Explanation
323/5 = (321/5 )3 = 23 = 8 .
🚩 Report
Simplify: \( (\sqrt{16})^3 \)
Explanation
\( (\sqrt{16})^3 = 4^3 = 64 \).
🚩 Report
Write as a radical expression: y2/3
A
2y3
B
\( \sqrt{y^3} \)
C
\( \sqrt[3]{y^2} \)
D
3y2
Explanation
Using the rule \( y^{m/n} = \sqrt[n]{y^m} \), the expression y2/3 becomes \( \sqrt[3]{y^2} \).
🚩 Report
Simplify the following expression:
85/3
Explanation
85/3 = (81/3 )5 = 25 = 32 .
🚩 Report
Simplify:
64-2/3
Explanation
\( 64^{-2/3} = \frac{1}{64^{2/3}} = \frac{1}{(\sqrt[3]{64})^2} = \frac{1}{4^2} = \frac{1}{16} \).
🚩 Report
Solve the equation: \( \sqrt{q+1} = 2 \)
Explanation
Squaring both sides gives q+1 = 4 , so q = 3 .
🚩 Report
What is the first step to solve: \( p = \sqrt{4p+8} - 3 \)?
A
square both sides
B
subtract 8 on both sides
C
add 3 on both sides
D
divide by 4 on both sides
Explanation
To solve a radical equation, you must first isolate the radical. Adding 3 to both sides results in \( p+3 = \sqrt{4p+8} \).
🚩 Report
What is the first step to solve:
(41k-31)1/4 = 5 ?
A
divide by 41 on both sides
B
add 31 on both sides
C
multiply both sides by 4
D
raise both sides to the 4th power
Explanation
The expression is already isolated. To eliminate the exponent of 1/4 , raise both sides to the power of 4.
🚩 Report
Solve for x: 6x5 = -192
Explanation
x5 = -192/6 = -32 . The fifth root of -32 is -2.
🚩 Report
Simplify. Your answer should contain only positive exponents:
yx1/3 × xy3/2
A
x4/3 y5/2
B
x2/3 y1/2
C
x1/3 y3/2
D
x2/3 y4/3
Explanation
Multiply the variables by adding their exponents: x1/3+1 = x4/3 and y1+3/2 = y5/2 .
🚩 Report
Simplify. Your answer should contain only positive exponents: \( \frac{a^2 b^0}{3a^4} \)
A
1
B
\( \frac{1}{3a^2} \)
C
\( \frac{a^2}{3a^4} \)
D
\( \frac{1}{3a^4} \)
Explanation
Since b0 = 1 , the expression is \( \frac{a^2}{3a^4} \). Subtracting exponents gives \( \frac{1}{3a^{4-2}} = \frac{1}{3a^2} \).
🚩 Report
Simplify. Your answer should contain only positive exponents:
(x0 y1/3 )3/2 × x0
Explanation
Since x0 = 1 , the expression simplifies to (y1/3 )3/2 . Multiply exponents: 1/3 × 3/2 = 1/2 .
🚩 Report
Find the indicated real nth root(s) of a: n = 6, a = -729
A
3
B
-3
C
\( \pm 3 \)
D
no real roots
Explanation
An even root (n=6) of a negative number has no real solution.
🚩 Report
Find the indicated real nth root(s) of a: n = 4, a = 256
A
-4
B
4
C
\( \pm 4 \)
D
no real roots
Explanation
For an even index (n=4) and a positive radicand, there are two real roots: positive and negative. Since 44 = 256 , the roots are \( \pm 4 \).
🚩 Report
Match the equivalent expression: \( 1/(\sqrt[4]{5}) \)
A
54/3
B
-51/4
C
53/4
D
5-1/4
Explanation
The fourth root is 51/4 . Because it is in the denominator, the exponent becomes negative: 5-1/4 .
🚩 Report
Match the equivalent expression: \( (\sqrt[4]{5})^3 \)
A
54/3
B
-51/4
C
53/4
D
5-1/4
Explanation
\( \sqrt[4]{5} \) is 51/4 . Raising this to the 3rd power gives (51/4 )3 = 53/4 .
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