Exponents with Like Bases

Q.

Express $625$ as a power of $5$.

Q.

Find each of the following products:

(a) 3 × (− 1) (b) (− 1) × 225

(c) (− 21) × (− 30) (d) (− 316) × (− 1)

(e) (− 15) × 0 × (−18) (f) (− 12) × (− 11) × (10)

(g) 9 × (− 3) × (− 6) (h) (− 18) × (− 5) × (− 4)

(i) (− 1) × (−2) × (− 3) × 4

(j) (− 3) × (− 6) × (− 2) × (− 1)

Q.

Two sets $A$ and $B$ are as under :

$A=\left\{(a,b)\in R\times R:|a-5|<1,\left|b-5\right|<1\right\}$

$B=\left\{(a,b)\in R\times R:4{(a-6)}^2+9{(b-5)}^2\le 36\right\}$ Then:

1. $A\cap B=\varphi$(an empty set)

2. neither $A\subset B$ nor $B\subset A$

3. $B\subset A$

4. $A\subset B$

Q.

Question 84 (a)

Simplify and express each fo the following in exponential form.
$[{\left(\frac{3}{7}\right)}^4\times {\left(\frac{3}{7}\right)}^5]÷{\left(\frac{3}{7}\right)}^7$

Q.

Question 54

$x^m\times x^n=x^{m+n},$ where $x$ is a non-zero rational number and $m,n$ are positive integers.

Q.

The value of ${\int }_0^{\infty }\frac{1}{\left[\left(x^2+4\right)\left(x^2+9\right)\right]}dx$ is

1. $\frac{\mathrm{\pi }}{60}$

2. $\frac{\pi }{20}$

3. $\frac{\pi }{40}$

4. $\frac{\pi }{80}$

Q.

The standard form of the number $12345$ is

(a) $1234.5\times {10}^1$
(b) $123.45\times {10}^2$
(a) $12.345\times {10}^3$
​​​​​​​(a) $1.2345\times {10}^4$

Q. Question 96:

A googal is the number 1 followed by 100 zeroes.
(a) How is a googol written as a power?
(b) How is a googol times a googol written as a power?

Q.

Question 85 (c)

Evaluate
$\frac{125\times 5^2\times a^7}{{10}^3\times a^4}$

Q.

Question 90:

If $2^{n+2}-2^{n+1}+2^n=C\times 2^n,thenfindC.$

Q.

Question 84 (b)

Simplify $[{\left(\frac{7}{11}\right)}^5÷{\left(\frac{7}{11}\right)}^2]\times {\left(\frac{7}{11}\right)}^2$

Q. Simplify [4 MARKS]

${\left(\frac{5}{8}\right)}^{-7}\times {\left(\frac{8}{5}\right)}^{-5}$

Q.

Question 7

The value of $\frac{{10}^{22}+{10}^{20}}{{10}^{20}}$ is
(a) 10
(b) ${10}^{42}$
(c) 101
(d) ${10}^{22}$

Q.

$\frac{1}{1!(n-1)!}+\frac{1}{3!(n-3)!}+\frac{1}{5!(n-5)!}+....$

1. $\frac{2^{n-4}}{n!}$for even values of n only

2. $\left(\frac{2^{n-4}+1}{n!}\right)-1$ for even values of n only

3. $\frac{1}{n!}\left(2^{n-1}\right)$ for all values of n

4. None of these

Q.

Question 85 (f)

Evaluate
$\frac{{15}^4\times {18}^3}{3^3\times 5^2\times {12}^2}$

Q.

Question 15

Which of the following is not equal to 1?
(a)$\frac{2^3\times 3^2}{4\times 18}$
(b)$\left[\right(-2{)}^3\times (-2{)}^4]÷(-2{)}^7$
(c)$\frac{3^0\times 5^3}{5\times 25}$
(d)$\frac{2^4}{(7^0+3^0{)}^3}$

Q.

If $x$ be any non-zero integer and $m,n$ be negative integers, then $x^m\times x^n$ is equal to:

1. $x^m$

2. $x^{m+n}$

3. $x^n$

4. $x^{m-n}$

Q.

Question 84 (d)

Simplify and express each fo the following exponential form.
$\left(\frac{a^6}{a^4}\right)\times a^5\times a^0$

Q.

Let $A=\left\{x|x\le 9,x\in N\right\}$. Let $B=\left\{a,b,c\right\}$ be the subset of $A$ where $\left(a+b+c\right)$ is a multiple of $3$. What is the largest possible number of subsets like $B$?

1. $12$

2. $21$

3. $27$

4. $30$

Q.

Question 81 (f)

Express each of the following in single exponential form.
$(-11{)}^2\times (-2{)}^2$

Q. $5^{x-3}\times 3^{2x-8}=225$

Q.

What is the value of ${100}^0$?

Q.

Find the number from each of the following expanded forms: ${(9\times 10)}^5+{(2\times 10)}^2+{(3\times 10)}^1$.

Q.

If $n$ is the number of ways of dividing $2n$ people into $n$ couples then

1. $2^{n}N=\left(2n\right)!$

2. $N(n!)=(1.3.5\dots (2n-1\left)\right)$

3. $N={}^{2n}{C}_n$

4. None of these

Q. Question 2(ix)
Simplify and express each of the following in exponential form:
(ix) $\frac{2^8\times a^5}{4^3\times a^3}$

Q. Question 2(iii)
Simplify and express each of the following in exponential form:
(iii) ${25}^4÷5^3$

Q.

Find the value of $x$, if ${\left(\sqrt{\frac{3}{5}}\right)}^{1-2x}=\frac{125}{27}$

Q. $a^x\times a^y\times a^z$ can be written as

1. $a^{xyz}$
2. $a^{x\times y+z}$
3. $a^{x+y+z}$
4. $a^{x+y\times z}$

Q.

Question 81 (e)

Express each of the following in single exponential form.
$(-3{)}^3\times (-10{)}^3$

Q.

Question 84 (f)

Simplify and express each of the following in exponential form.
$(5^{15}÷5^{10})\times 5^5$