Question paper, Set 8

$\displaystyle \left(\frac{5}{6}\right)^{3}$ means $(5, 6)$ multiplied by itself 3 times.

$\displaystyle \left(\frac{5}{6}\right)^{3} = \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6}$

Answer: $\displaystyle \frac{5}{6} \times \frac{5}{6} \times \frac{5}{6}$

$4^{5} = 4 \times 4 \times 4 \times 4 \times 4 = 1024$

$5^{4} = 5 \times 5 \times 5 \times 5 = 625$

$1024 > 625$, so $4^{5}$ is greater.

Answer: $4^{5}$, since $1024 > 625$

Divide repeatedly by $4$: $4096, 1024, 256, 64, 16, 4$.

$4096 = 4 \times 4 \times 4 \times 4 \times 4 \times 4$

$= 4^{6}$

Answer: $4096 = 4^{6}$

Using $a^m \times a^n = a^{m+n}$

$12^{7} \times 12^{9} \times 12^{14} = 12^{7+9+14}$

$= 12^{30}$

Answer: $12^{30}$

Using $(a^m)^n = a^{m \times n}$

$(y^{5})^{7} = y^{5 \times 7}$

$= y^{35}$

Answer: $y^{35}$

Using $a^n \times b^n = (a \times b)^n$

$8^{2} \times 5^{2} = (8 \times 5)^{2}$

$= (40)^{2} = 40 \times 40$

$= 1600$

Answer: $(40)^{2} = 1600$

$9^{0} = 1$

$9^{0} - 1 = 1 - 1 = 0$

Answer: $0$

Divide repeatedly by $9$: $729, 81, 9$, so $729 = 9^{3}$.

$\displaystyle \frac{1}{729} = \frac{1}{9^{3}}$

$= 9^{-3}$ (since $\displaystyle \frac{1}{a^n} = a^{-n}$)

Answer: $9^{-3}$

Using $\displaystyle \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$

$\displaystyle \left(\frac{2}{13}\right)^{-7} = \left(\frac{13}{2}\right)^{7}$

Answer: $\displaystyle \left(\frac{13}{2}\right)^{7}$

$40$ is even, so $(-1)^{40} = 1$; $52$ is even, so $(-1)^{52} = 1$.

$(-1)^{40} \times (-1)^{52} = 1 \times 1 = 1$

Answer: $1$

Using $(-a)^n = -a^n \text{ if } n \text{ is odd}, \; (-a)^n = a^n \text{ if } n \text{ is even}$

$(-10)^{3} = -10^{3} = -1000$ ($3$ is odd)

$(-10)^{4} = 10^{4} = 10{,}000$ ($4$ is even)

Answer: $(-10)^{3} = -1000$, $(-10)^{4} = 10{,}000$

Given, $10^{x+3} = 1{,}00{,}000$

Dividing $1{,}00{,}000$ repeatedly by $10$: $1{,}00{,}000, 10{,}000, 1000, 100, 10$, so $1{,}00{,}000 = 10^{5}$.

$\Rightarrow 10^{x+3} = 10^{5}$

Using $a^m = a^n \Rightarrow m = n$

$\Rightarrow x+3 = 5$

$\Rightarrow x = 5 - 3$

$\Rightarrow x = 2$

Answer: $x = 2$

Using $\displaystyle a^{m-n} = \frac{a^m}{a^n}$

$\displaystyle 4^{x-3} = \frac{4^{x}}{4^{3}}$

$\displaystyle = \frac{4^{x}}{64}$

Answer: $\displaystyle \frac{4^{x}}{64}$

Using $(ab)^n = a^n \times b^n$ and $(a^m)^n = a^{m \times n}$

$(5t^{4})^{3} = 5^{3} \times (t^{4})^{3}$

$= 125 \times t^{12} = 125t^{12}$

Answer: $125t^{12}$

Using $(ab)^n = a^n \times b^n$ and $(a^m)^n = a^{m \times n}$

$(m^{3}n)^{2} \times (mn^{2})^{2} = m^{6}n^{2} \times m^{2}n^{4}$

Using $a^m \times a^n = a^{m+n}$: $= m^{8}n^{6}$

Answer: $m^{8}n^{6}$

Using $\displaystyle a^{-n} = \frac{1}{a^n}$

$\displaystyle (-5)^{-3} = \frac{1}{(-5)^{3}}$

$\displaystyle = \frac{1}{-125} = -\frac{1}{125}$ ($3$ is odd)

Answer: $\displaystyle -\frac{1}{125}$

(i) Given, $\displaystyle 4^{2y-1} = \frac{1}{16}$

$\displaystyle \frac{1}{16} = \frac{1}{4^{2}} = 4^{-2}$ (dividing $16$ repeatedly by $4$: $16, 4$)

$\Rightarrow 4^{2y-1} = 4^{-2}$

Using $a^m = a^n \Rightarrow m = n$

$\Rightarrow 2y-1 = -2$

$\Rightarrow 2y = -2 + 1$

$\Rightarrow 2y = -1$

$\displaystyle \Rightarrow y = -\frac{1}{2}$

(ii) Given, $(-5)^{x+1} \times (-5)^{2} = (-5)^{6}$

$\Rightarrow (-5)^{x+1+2} = (-5)^{6}$ ($a^m \times a^n = a^{m+n}$)

Using $a^m = a^n \Rightarrow m = n$

$\Rightarrow x+3 = 6$

$\Rightarrow x = 6 - 3$

$\Rightarrow x = 3$

Answer: (i) $\displaystyle y = -\frac{1}{2}$ (ii) $x = 3$

(i) $\displaystyle \left(5^{-6} \div 5^{-8}\right) \times 5^{1} = \frac{5^{-6}}{5^{-8}} \times 5^{1}$

$= 5^{-6+8} \times 5^{1} = 5^{2} \times 5^{1}$

$= 5^{2+1} = 5^{3}$

$= 125$

(ii) Using $a^n \times b^n = (a \times b)^n$

$\displaystyle 4^{2} \times \left(\frac{7}{4}\right)^{2} = \left(4 \times \frac{7}{4}\right)^{2}$

$= 7^{2} = 49$

Answer: (i) $125$ (ii) $49$

(i) Using $a^m \times a^n = a^{m+n}$

$\displaystyle 10^{\frac{5}{2}} \times 10^{\frac{1}{2}} = 10^{\frac{5}{2} + \frac{1}{2}}$

$\displaystyle = 10^{\frac{6}{2}} = 10^{3}$

$= 1000$

(ii) Using $\displaystyle \frac{a^m}{a^n} = a^{m-n}$

$\displaystyle 7^{\frac{9}{2}} \div 7^{\frac{3}{2}} = 7^{\frac{9}{2} - \frac{3}{2}}$

$\displaystyle = 7^{\frac{6}{2}} = 7^{3}$

$= 343$

Answer: (i) $1000$ (ii) $343$

(i) $\displaystyle \left(5^{2} \div 5^{5}\right)^{2} \times 5^{4} = \left(\frac{5^{2}}{5^{5}}\right)^{2} \times 5^{4}$

$= (5^{2-5})^{2} \times 5^{4} = (5^{-3})^{2} \times 5^{4}$

$= 5^{-6} \times 5^{4}$ ($(a^m)^n = a^{m \times n}$)

$= 5^{-6+4} = 5^{-2}$

$\displaystyle = \frac{1}{5^{2}} = \frac{1}{25}$

(ii) $\displaystyle \left(2^{-1} - 5^{-1}\right) \div 4^{-1} = \left(\frac{1}{2} - \frac{1}{5}\right) \div \frac{1}{4}$

$\displaystyle = \frac{5 - 2}{10} \div \frac{1}{4} = \frac{3}{10} \div \frac{1}{4}$

$\displaystyle = \frac{3}{10} \times 4 = \frac{6}{5}$

Answer: (i) $\displaystyle \frac{1}{25}$ (ii) $\displaystyle \frac{6}{5}$