1003118108
Level:
C
The value of \( \sqrt{5\left(222^2+111^2\right)} \) is:
\( 555 \)
\( 111\sqrt5 \)
\( \sqrt{15} \)
\( \sqrt{1110^2-555^2} \)
Expressions with exponents and radicals
1003118107
Level:
B
The number \( \left(\frac{27^{-4}\cdot8^{-4}}{16^{-2}\cdot9^{-5}}\right)^{-3} \) equals:
\( 12^{6} \)
\( 6^6 \)
\( 6^{12} \)
\( \frac1{3^6\cdot2^{12}} \)
1003118105
Level:
C
The number \( \frac1{\left(2-\sqrt3\right)^3} \) is equal to:
\( 26+15\sqrt3 \)
\( 27+24\sqrt3 \)
\( 14+7\sqrt3 \)
\( 27+30\sqrt3 \)
1003118104
Level:
C
Simplify \(\sqrt[4]{\left(\sqrt3-\sqrt2\right)^4}+\sqrt[4]{\left(\sqrt2-\sqrt5\right)^4}+\sqrt[3]{\left(\sqrt3-\sqrt5\right)^3} \).
\( 2\sqrt3-2\sqrt2 \)
\( 2\sqrt5 - 2\sqrt2 \)
\( 2\sqrt3-2\sqrt5 \)
\( 2\sqrt5-2\sqrt3 \)
1003118103
Level:
B
Express \( \frac{\sqrt[3]5\sqrt[6]5}{3\cdot25^3+2\cdot125^2} \) as a power of \( 5 \).
\( 5^{-\frac{13}2} \)
\( 5^{-5} \)
\( 5^{-6} \)
\( 5^{-\frac{11}2} \)
1003118101
Level:
C
Which of the following inequalities describes a correct relationship between the numbers \( a \), \( b \), \( c \) if \( a=4^{2^4} \), \( b=3^{4^3} \), \( c=2^{3^4} \)? (Hint: Barring parentheses, \( x^{y^z} \) is equivalent to \( x^{\left(y^z\right)} \).)
\( a < c < b \)
\( b < a < c \)
\( a < b < c \)
\( b < c < a \)
1003134510
Level:
C
Complete the following sentence to obtain a true statement. The number \( \left( \sqrt3 \right)^{\sqrt2^{\sqrt[3]{64}}} \) is ...
a rational number less than \( 10 \).
an irrational number less than \( 10 \).
an integer greater than \( 10 \).
a rational number greater than \( 10 \).
1003134509
Level:
B
Let \( a=4^{1.5} \) and \( b=0.125^{-\frac13} \). Choose the true statement.
\( a=4b \)
\( a=\frac12b \)
\( a=2b \)
\( a < b \)
1003134508
Level:
C
The number \( \left(\sqrt2-1\right)^6 \) is equal to:
\( 99-70\sqrt2 \)
\( 5\sqrt2 - 7 \)
\( 49-35\sqrt2 \)
\( 34-24\sqrt2 \)
1003134507
Level:
B
In the expansion of \( \left( 2\sqrt3x+4y\right)^3 \) the coefficient of \( xy^2 \) is:
\( 96\sqrt3 \)
\( 32\sqrt3 \)
\( 48 \)
\( 144 \)