Arithmetic aptitude :: Surds and Indices :: General Questions

• Surds and Indices - General Questions
• Surds and Indices - Important Formulas and Facts

Laws of Indices :

1.  $a^{m} \times a^{n}= a^{m+n}$
2.  $\frac{a^{m}}{a^{n}}= a^{m-n}$
3.  $(a^{m})^{n}=a^{mn}$
4.  $(ab)^{n} = a^{n}b^{n}$
5.  $\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}$
6.  $a^{0}=1$

Surds:

When we can't simplify a number to remove a square root (or cube root etc) then it is called a surd.

Rational Number:

A real number that can be written as a simple fraction then it is called rational number

e.g. 1.5 = $\frac{3}{2}$

Irrational Number:

A real number that can't be written as a simple fraction is called Irrational number.

e.g. $\pi$ = 3.1415926535897932384626433832795

Let a be rational number and n be a positive integer such that $a^{\frac{1}{n}}=\sqrt[n]{a}$ is irrational. Then $\sqrt[n]{a}$ is called surd of order n.

Laws of Surds

1. $\sqrt[n]{a}=a^{\frac{1}{n}}$
2. $\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$
3. $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$
4. $(\sqrt[n]{a})^{n}= a$
5. $\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$
6. $(\sqrt[n]{a})^{m} = \sqrt[n]{a^{m}}$

If $\frac{9^{n} \times 3^{5} \times \left(27\right)^{3}}{3 \times \left(81\right)^{4}}=27$, then the value of n is:

Answer:

Explanation:

$\frac{9^{n} \times 3^{5} \times \left(27\right)^{3}}{3 \times \left(81\right)^{4}}=27$ $\Leftrightarrow \frac{\left(3^{2}\right)^{n} \times 3^{5} \times \left(3^{3}\right)^{3}}{3 \times \left(3\right)^{4\times 4}}=3^{3}$ $\Leftrightarrow \frac{\left(3\right)^{2n} \times 3^{5} \times \left(3\right)^{3\times 3}}{3 \times \left(3\right)^{4\times 4}}=3^{3}$ $\Leftrightarrow\frac{3^{2n+5+9}}{3 \times 3^{16}}=3^{3}$ $\Leftrightarrow\frac{3^{2n+14}}{ 3^{17}}=3^{3}$ $\Leftrightarrow 3^{\left(2n+14-17\right)}=3^{3}$ $\Leftrightarrow 3^{\left(2n-3\right)}=3^{3}$ $2n-3 = 3 \Leftrightarrow 2n=6$ $\Leftrightarrow n=3$

If $\left(\sqrt{3}\right)^{5} \times 9^{2}=3^{n}\times 3\sqrt{3}$, then the value of n is:

Answer:

Explanation:

$\left(\sqrt{3}\right)^{5} \times 9^{2}=3^{n}\times 3\sqrt{3}$ $\Leftrightarrow \left(3^{\frac{1}{2}}\right)^{5} \times \left(3^{2}\right)^{2}$ $ = 3^{n}\times3 \times 3^{\frac{1}{2}}$ $\Leftrightarrow 3^{\left(\frac{1}{2}\times 5\right)}\times 3^{\left(2\times 2\right)}$ $=3^{\left(n+1+\frac{1}{2}\right)}$ $\Leftrightarrow3^{\left(\frac{5}{2}+4\right)}$ $=3^{\left(n+\frac{3}{2}\right)}$ $\Leftrightarrow n+\frac{3}{2}=\frac{13}{2}$ $\Leftrightarrow n=\left(\frac{13}{2}-\frac{3}{2}\right)$ $=\frac{10}{2}=5$

If $\sqrt{2^{n}}=64$ then the value of n is:

Answer:

Explanation:

$\sqrt{2^{n}}=64$ $\Leftrightarrow \left(2^{n}\right)^{\frac{1}{2}}$ $=2^{6}$ $\Leftrightarrow 2^{\frac{n}{2}}=2^{6}$ $\Leftrightarrow \frac{n}{2}=6$ $\Leftrightarrow n=12$

If $5\sqrt{5}\times 5^{3}+5^{-\frac{3}{2}}$ $=5^{a+2}$, then the value of a is:

Answer:

Explanation:

$5\sqrt{5}\times 5^{3}+5^{-\frac{3}{2}}$ $=5^{a+2}$ $\Leftrightarrow \frac{5 \times 5^{\frac{1}{2}} \times 5^{3}}{5^{-\frac{3}{2}}}$ $\Leftrightarrow 5^{\left(1+\frac{1}{2}+3+\frac{3}{2}\right)}$ $=5^{a+2}$ $\Leftrightarrow 5^{6}$ $=5^{a+2}$ $\Leftrightarrow a+2=6$ $\Leftrightarrow a=4$

If $2^{2n-1}=\frac{1}{8^{n-3}}$, then the value of n is:

Answer:

Explanation:

$2^{2n-1}=\frac{1}{8^{n-3}}$ $\Leftrightarrow 2^{2n-1}=\frac{1}{\left(2^{3}\right)^{n-3}}$ $=\frac{1}{2^{3}\left(^{n-3}\right)}$ $=\frac{1}{2\left(^{3n-9}\right)}$ $=2^\left({9-3n}\right)$ $\Leftrightarrow 2n-1$ $=9-3n$ $\Leftrightarrow 5n=10$ $\Leftrightarrow n=2$

• Surds and Indices - General Questions