# Surds and Indices Questions for CAT PDF

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Surds and Indices are some of the basics of number system theory. The questions may include simplification of equations to find the simplified equations. Practice more questions on Surds and indices for CAT.

Surds and Indices Questions for CAT PDF:

Question 1:  If $3^{2x + 3} – 244*3^x = -9$, then which of the following statements is true?

a) ‘x’ is a positive number
b) ‘x’ is a negative number
c) ‘x’ can be either a positive number or a negative number
d) None of the above

Question 2: If x = -0.5, then which of the following has the smallest value?

a) $2^{1/x}$
b) $1/x$
c) $1/x^2$
d) $2^x$
e)$1/\sqrt{-x}$

Question 3: Which of the following is true?

a) $7^{(3^2)} = (7^3)^2$
b) $7^{(3^2)} > (7^3)^2$
c) $7^{(3^2)} < (7^3)^2$
d) None of these

Question 4: Which among $2^{1/2}, 3^{1/3}, 4^{1/4}, 6^{1/6}$, and $12^{1/12}$ is the largest?

a) $2^{1/2}$
b) $3^{1/3}$
c) $4^{1/4}$
d) $6^{1/6}$
e)$12^{1/12}$

Question 5: $2^{73}-2^{72}-2^{71}$ is the same as

a) $2^{69}$
b) $2^{70}$
c) $2^{71}$
d) $2^{72}$

Solutions for Surds and Indices Questions for CAT PDF:

Solutions:

The equation can be written as follow:
$3^{2x} * 3^3 – 244 * 3^x + 9 = 0$
Let $3^x = t$
=> $27t^2 – 244t + 9 = 0$
=> $27t^2 – 243t – t + 9 = 0$
=> $27t(t – 9) – 1(t – 9) = 0$
=> t = 9 or t = 1/27
=> $3^x = 9 or 3^x = 1/27$
=> $3^x = 3^2 or 3^x = 3^{-3}$
So, x = 2 or x = -3
So, ‘x’ can be either a positive number or a negative number. Option c) is the correct answer

$2^p$ is always positive
$x^2$ is always non negative.
$1/\sqrt{-x}$ is always positive.
$\frac{1}{x}$ is negative when x is negative.
In this case, x is negative => $\frac{1}{x}$ is smallest.

$7^{(3^2)} = 7^9$
$(7^3)^2 = 7^6$
So $7^{(3^2)} > (7^3)^2$

Make the power equal and compare the denominators.
$2^{1/2}$ can be written as $64^{1/12}$
$3^{1/3}$ can be written as $81^{1/12}$
$4^{1/4}$ can be written as $64^{1/12}$
$6^{1/6}$ can be written as $36^{1/12}$
Among these, $81^{1/12}$ is the greatest => $3^{1/3}$ is the greatest.

$2^{71} (2^2 – 2^1 – 1)$
$2^{71} (4-2-1)$
$2^{71}$