## Surds and Indices Questions and Answers Part-8

1. $${\left( {\frac{{{x^b}}}{{{x^c}}}} \right)^{\left( {b + c - a} \right)}}.$$   $${\left( {\frac{{{x^c}}}{{{x^a}}}} \right)^{\left( {c + a - b} \right)}}.$$   $${\left( {\frac{{{x^a}}}{{{x^b}}}} \right)^{\left( {a + b - c} \right)}} = ?$$
a) xabc
b) 1
c) xab+bc+ca
d) xa+b+c

Explanation:
$${x^{\left( {b - c} \right)\left( {b + c - a} \right)}}.{x^{\left( {c - a} \right)\left( {c + a - b} \right)}}.{x^{\left( {a - b} \right)\left( {a + b - c} \right)}}$$
$$= {x^{\left( {b - c} \right)\left( {b + c} \right) - a\left( {b - c} \right)}}.$$    $${x^{\left( {c - a} \right)\left( {c + a} \right) - b\left( {c - a} \right)}}.$$   $${x^{\left( {a - b} \right)\left( {a + b} \right) - c\left( {a - b} \right)}}$$
\eqalign{ & = {x^{\left( {{b^2} - {c^2} + {c^2} - {a^2} + {a^2} - {b^2}} \right)}}.{x^{ - a\left( {b - c} \right) - b\left( {c - a} \right) - c\left( {a - b} \right)}} \cr & = \left( {{x^0} \times {x^0}} \right) \cr & = \left( {1 \times 1} \right) \cr & = 1 \cr}

2.If 2n-1 + 2n+1 = 320, then the value of n is = ?
a) 6
b) 8
c) 5
d) 7

Explanation:
\eqalign{ & {\text{ }}{{\text{2}}^{n - 1}}{\text{ + }}{{\text{2}}^{n + 1}}{\text{ = 320}} \cr & {\text{ }}{{\text{2}}^{n - 1}}\left( {1 + {2^2}} \right){\text{ = 320}} \cr & {\text{ }}{{\text{2}}^{n - 1}} \times {\text{5 = 320}} \cr & {\text{ }}{{\text{2}}^{n - 1}}{\text{ = }}\frac{{320}}{5}{\text{ = 64}} \cr & {\left( 2 \right)^{n - 1}} = {\left( 2 \right)^6} \cr & n - 1 = 6 \cr & n = 7 \cr}

3. 461 + 462 + 463 + 464 is divided by = ?
a) 17
b) 3
c) 11
d) 13

Explanation:
\eqalign{ & {4^{61}} + {4^{62}} + {4^{63}} + {4^{64}} \cr & = {4^{61}}\left( {{4^0} + {4^1} + {4^2} + {4^3}} \right) \cr & = {4^{61}} \times 85 \cr}
Now check with option 85 is divisible by 17.

4. The value of $${\left( {{x^{\frac{{b + c}}{{c - a}}}}} \right)^{\frac{1}{{a - b}}}}{\text{.}}$$  $${\left( {{x^{\frac{{c + a}}{{a - b}}}}} \right)^{\frac{1}{{b - c}}}}.$$  $${\left( {{x^{\frac{{a + b}}{{b - c}}}}} \right)^{\frac{1}{{c - a}}}}{\text{ is = ?}}$$
a) 1
b) a
c) b
d) c

Explanation:
\eqalign{ & {x^{\frac{{b + c}}{{\left( {a - b} \right)\left( {c - a} \right)}}}}.{x^{\frac{{c + a}}{{\left( {a - b} \right)\left( {b - c} \right)}}}}.{x^{\frac{{a + b}}{{\left( {b - c} \right)\left( {c - a} \right)}}}} \cr & = {x^{\frac{{\left( {b + c} \right)\left( {b - c} \right) + \left( {c + a} \right)\left( {c - a} \right) + \left( {a + b} \right)\left( {a - b} \right)}}{{\left( {a - b} \right)\left( {b - c} \right)\left( {c - a} \right)}}}} \cr & = {x^{\frac{{\left( {{b^2} - {c^2}} \right) + \left( {{c^2} - {a^2}} \right) + \left( {{a^2} - {b^2}} \right)}}{{\left( {a - b} \right)\left( {b - c} \right)\left( {c - a} \right)}}}} \cr & = {x^0} \cr & = 1 \cr}

5. If ax = b, by = c and cz = a, then the value of xyz is =
a) 0
b) 1
c) $$\frac{1}{{{\text{abc}}}}$$
d) abc

Explanation:
\eqalign{ & {a^1} \cr & = {c^z} \cr & = {\left( {{b^y}} \right)^z} \cr & = {b^{yz}} \cr & = {\left( {{a^x}} \right)^{yz}} \cr & = {a^{xyz}} \cr & \Rightarrow xyz = 1 \cr}

6.(0.04)2 ÷ (0.008) × (0.2)6 = (0.2)?
a) 5
b) 6
c) 8
d) None of these

Explanation:
\eqalign{ & {\text{Let }}{\left( {0.04} \right)^2} \div \left( {0.008} \right) \times {\left( {0.2} \right)^6} = {\left( {0.2} \right)^x} \cr & {\text{Then,}}{\left( {0.2} \right)^x} = {\left[ {{{\left( {0.2} \right)}^2}} \right]^2} \div {\left( {0.2} \right)^3} \times {\left( {0.2} \right)^6} \cr & {\left( {0.2} \right)^x} = {\left( {0.2} \right)^{\left( {2 \times 2} \right)}} \div {\left( {0.2} \right)^3} \times {\left( {0.2} \right)^6} \cr & {\left( {0.2} \right)^x} = {\left( {0.2} \right)^4} \div {\left( {0.2} \right)^3} \times {\left( {0.2} \right)^6} \cr & {\left( {0.2} \right)^x} = {\left( {0.2} \right)^{\left( {4 - 3 + 6} \right)}} \cr & {\left( {0.2} \right)^x} = {\left( {0.2} \right)^7} \cr & x = 7 \cr}

7. The value of $$\frac{{{{\left( {243} \right)}^{0.13}} \times {{\left( {243} \right)}^{0.07}}}}{{{{\left( 7 \right)}^{0.25}} \times {{\left( {49} \right)}^{0.075}} \times {{\left( {343} \right)}^{0.2}}}}$$      is = ?
a) $$\frac{3}{7}$$
b) $$\frac{7}{3}$$
c) $${\text{1}}\frac{3}{7}$$
d) $${\text{2}}\frac{2}{7}$$

Explanation:
\eqalign{ & \frac{{{{\left( {243} \right)}^{0.13}} \times {{\left( {243} \right)}^{0.07}}}}{{{{\left( 7 \right)}^{0.25}} \times {{\left( {49} \right)}^{0.075}} \times {{\left( {343} \right)}^{0.2}}}} \cr & = \frac{{{{\left( {243} \right)}^{\left( {0.13 + 0.07} \right)}}}}{{{{\left( 7 \right)}^{0.25}} \times {{\left( {{7^2}} \right)}^{0.075}} \times {{\left( {{7^3}} \right)}^{0.2}}}} \cr & = \frac{{{{\left( {243} \right)}^{0.2}}}}{{{{\left( 7 \right)}^{0.25}} \times {{\left( 7 \right)}^{\left( {2 \times 0.075} \right)}} \times {{\left( 7 \right)}^{\left( {3 \times 0.2} \right)}}}} \cr & = \frac{{{{\left( {{3^5}} \right)}^{0.02}}}}{{{{\left( 7 \right)}^{0.25}} \times {{\left( 7 \right)}^{0.15}} \times {{\left( 7 \right)}^{0.6}}}} \cr & = \frac{{{{\left( 3 \right)}^{\left( {5 \times 0.2} \right)}}}}{{{{\left( 7 \right)}^{\left( {0.25 + 0.15 + 0.6} \right)}}}} \cr & = \frac{{{3^1}}}{{{7^1}}} \cr & = \frac{3}{7} \cr}

8. $$\frac{{{2^{n + 4}} - 2 \times {2^n}}}{{2 \times {2^{\left( {n + 3} \right)}}}} + {2^{ - 3}}$$     is equal to = ?
a) 2(n+1)
b) $$\left( {\frac{9}{8} - {2^n}} \right)$$
c) $$\left( { - {2^{n + 1}} + \frac{1}{8}} \right)$$
d) 1

Explanation:
\eqalign{ & \frac{{{2^{n + 4}} - 2 \times {2^n}}}{{2 \times {2^{\left( {n + 3} \right)}}}} + {2^{ - 3}} \cr & = \frac{{{2^{n + 4}} - {2^{n + 1}}}}{{{2^{\left( {n + 4} \right)}}}} + \frac{1}{{{2^3}}} \cr & = \frac{{{2^{n + 1}}\left( {{2^3} - 1} \right)}}{{{2^{\left( {n + 4} \right)}}}} + \frac{1}{{{2^3}}} \cr & = \frac{{{2^{n + 1}} \times 7}}{{{2^{n + 1}} \times {2^3}}} + \frac{1}{{{2^3}}} \cr & = \left( {\frac{7}{8} + \frac{1}{8}} \right) \cr & = \frac{8}{8} \cr & = 1 \cr}

9. $$\frac{{256 \times 256 - 144 \times 144}}{{112}}$$     is equal to = ?
a) 420
b) 400
c) 360
d) 320

\eqalign{ & \frac{{256 \times 256 - 144 \times 144}}{{112}} \cr & = \frac{{{{\left( {256} \right)}^2} - {{\left( {144} \right)}^2}}}{{112}} \cr & = \frac{{\left( {112} \right)\left( {400} \right)}}{{112}} \cr & = 400 \cr}
10. $$\sqrt {3\sqrt {3\sqrt {3........} } }$$    is equal to = ?
a) $$\sqrt 3$$
c) $${\text{2}}\sqrt 3$$
d) $${\text{3}}\sqrt 3$$
\eqalign{ & \Rightarrow \sqrt {n\sqrt {n\sqrt n } } ........\infty \cr & \Rightarrow {\text{So }}n{\text{ is answer }} \cr & \Rightarrow {\text{3}} \cr}