1003118008
Level:
C
Let \( a=\frac1{5+\sqrt3}+\frac1{44-22\sqrt3} \). Choose the correct statement about the number \( a \).
It is a rational number.
It is a positive integer.
It is an irrational number.
It is greater than \( 1 \).
Expressions with exponents and radicals
1003118010
Level:
C
The product of the number \( \sqrt{\sqrt2+1} \) and the multiplicative inverse of \( \sqrt{\sqrt2-1} \) is equal to:
\( 1+\sqrt2 \)
\( 2\sqrt2 \)
\( 1-\sqrt2 \)
\( 1 \)
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 \( x^{\left(y^z\right)} \).)
\( a < c < b \)
\( b < a < c \)
\( a < b < c \)
\( b < c < a \)
1003118104
Level:
C
Simplify \(\sqrt[4]{\left(\sqrt3-\sqrt2\right)^4}+\sqrt[4]{\left(\sqrt2-\sqrt5\right)^4}+\sqrt[3]{\left(\sqrt3-\sqrt5\right)^3} \).
\( 2\sqrt3-2\sqrt2 \)
\( 2\sqrt5 - 2\sqrt2 \)
\( 2\sqrt3-2\sqrt5 \)
\( 2\sqrt5-2\sqrt3 \)
1003118105
Level:
C
The number \( \frac1{\left(2-\sqrt3\right)^3} \) is equal to:
\( 26+15\sqrt3 \)
\( 27+24\sqrt3 \)
\( 14+7\sqrt3 \)
\( 27+30\sqrt3 \)
1003118108
Level:
C
The value of \( \sqrt{5\left(222^2+111^2\right)} \) is:
\( 555 \)
\( 111\sqrt5 \)
\( \sqrt{15} \)
\( \sqrt{1110^2-555^2} \)
1003118109
Level:
C
Which of the following numbers has the same unit digit as the number \( 555^{555} \)?
\( 10^{555}+5 \)
\( \frac{10^{555}}5 \)
\( 5\cdot10^{555} \)
\( 10^{555}+10^{555} \)
1003118110
Level:
C
Let \( \frac{u\sqrt5}{5-\sqrt5}=\frac{\frac{5+\sqrt5}{\sqrt5+1}}{1-\sqrt5} \). What is the reciprocal of \( u \)?
\( \frac1u=-\frac{\sqrt5}5 \)
\( \frac1u = \frac1{\sqrt5} \)
\( \frac1u = \frac5{\sqrt5} \)
\( \frac1u = -\sqrt5 \)
1003118201
Level:
C
Suppose that \( \sqrt[3]2\approx 1.25 \). Give the approximate value of \( \left(\frac1{81}\right)^{-\sqrt[3]2}\). Do not use a calculator.
\( 243 \)
\( 729 \)
\( \frac19 \)
\( 0 \)
1003118202
Level:
C
The approximate value of the number \( 10^{-0.8} \) is \( 0.158489 \). What is the approximate value of the number \( 10^{1.2} \)? Give your answer to three decimal places. Do not use a calculator.
\( 15.849 \)
\( 1.585 \)
\( 0.015 \)
\( 3.170 \)