Exponents with Unlike Bases and Same Exponent

Q.

Prove that : $(m+n{)}^{-1}(m^{-1}+n^{-1})=(mn{)}^{-1}$

Q.

Find the value of the following expression:

${625}^{-3/4}$

Q. Evaluate:$\sqrt[3]{31372}\times \sqrt[3]{31458}$

1. 64
2. 126
3. 124
4. 68

Q.

Simplify :

$\left(i\right)\frac{a^5{b}^2}{a^2{b}^{-3}}\left(ii\right)15y^8÷3y^3\left(iii\right)x^{10}{y}^6÷x^3{y}^{-2}\left(iv\right)5z^{16}÷15z^{-11}\left(v\right)\left(36x^2{)}^{\frac{1}{2}}\right(vi\left)\right(125x^{-3}{)}^{\frac{1}{3}}\left(vii\right)\left(2x^2{y}^{-3}{)}^{-2}\right(viii\left)\right(27x^{-3}{y}^6{)}^{\frac{2}{3}}\left(ix\right){(-2x^{\frac{2}{3}}{y}^{-\frac{3}{2}})}^6$

Q.

If $3^x$ = $\frac{1}{81}$ , the value of x is

__

Q.

The distance between Earth and Mars is 2.2253x${10}^8$ km. The distance between Mars and Saturn is $1.19666\times {10}^9$ km. A spaceship starts from Earth and flies to Saturn via Mars. Find the total distance covered by the spaceship in metres.

1. 1.41919x1012 m
2. 14191900000 m
3. 1419190000000 m
4. 1.41919x${10}^9$ km

Q.

Question 2(ii)
Simplify and express the result in power notation with positive exponent:

${\left(\frac{1}{2^3}\right)}^2$

Q.

Simplify $\frac{(\frac{1}{5}{)}^{-3}-(\frac{1}{3}{)}^{-4}}{(\frac{1}{4}{)}^{-3}}.$

1. $\frac{12}{13}$

2. $\frac{13}{12}$

3. $\frac{11}{16}$

4. $\frac{16}{11}$

Q.

Question 2(v)
Simplify and express the result in power notation with positive exponent:

$2^{-3}\times (-7{)}^{-3}$

Q.

Find the value of the following expression:

$\frac{8^{-2/3}}{{16}^{-3/4}}$

Q.

${\left(\right(2{)}^2)}^2$× ($3^3$ ÷$4^2$) ÷ $3^2$ is equal to

1. 3

2. 1

3. $\frac{12}{3}$
4. 27

Q.

The value of ${\left(-\frac{1}{2}\right)}^{-5}$ is:

1. $1$

2. $-2$

3. $-32$

4. $32$

Q.

Question 20

For non-zero numbers $a$ and $b$, ${\left(\frac{a}{b}\right)}^m÷{\left(\frac{a}{b}\right)}^n,$ where $m>n,$ is equal to:
(a)${\left(\frac{a}{b}\right)}^{mn}$
(b)${\left(\frac{a}{b}\right)}^{m+n}$
(c)${\left(\frac{a}{b}\right)}^{m-n}$
(d)${\left({\left(\frac{a}{b}\right)}^m\right)}^n$

Q.

Write a function rule for “the output is eight more than one-seventh of the input.”

Q.

Expand the following numbers using exponents:

a) 1029.63

b) 1296.249

Q.

For any two non-zero rational numbers a and b, $a^5\times b^5$ is equal to

1. $(a\times b{)}^{10}$

2. $(a\times b{)}^0$

3. $(a\times b{)}^5$

4. $(a\times b{)}^{25}$

Q.

Simplify: $\left(\frac{1}{6^{(-2)}}-\frac{1}{3^{(-3)}}\right)÷\frac{1}{5^{(-2)}}$

1. $\frac{5}{3}$

2. $\frac{9}{25}$

3. $\frac{3}{5}$

4. $\frac{25}{9}$

Q.

Expand $a^3{b}^2,a^2{b}^3,b^2{a}^3,b^3{a}^2$. Are they all same?

Q.

Write the following numbers in the expanded from with exponential notation:

$5,49,36,005$

Q.

Simplify using law(s) of exponents: $\sqrt{98}$

1. $6\sqrt{2}$

2. $7\sqrt{3}$

3. $7$

4. $7\sqrt{2}$

Q.

Find the value of $(\frac{1}{15}{)}^{-2}$ + $(\frac{1}{13}{)}^{-2}$ + $(\frac{1}{12}{)}^{-2}$.

1. 538

2. 269

3. 0.226

4. 41

Q. Simplify $\frac{(-2{)}^3\times (-2{)}^7}{3\times 4^6}.$

1. 2
2. $\frac{1}{12}$
3. $\frac{2}{3}$
4. $\frac{1}{6}$

Q.

Question 158(iii)

Find a single machine that will do the same job as the given hook-up.
$a(\times 5^{99})$ machine followed by a $\left(5^{-100}\right)$ machine.
Maya multiplied $(4^2\times 3^2)$ by thinking about stretching machines.

Use Maya's idea to multiply $5^3\times 2^3$.
Maya's idea is another product law of exponents.
Multiplying Expressions with the same Exponents
$a^c\times b^c=(a\times b{)}^c$
You can use this law with more than two expressions. If the exponents are the same, multiply the expressions by multiplying the bases and using the same exponent. e.g. $2^8\times 3^8\times 7^8=(2\times 3\times 7{)}^8={42}^8$.

Q.

The distance between Earth and Mars is 2.2253 $\times$ ${10}^8$ kms. The distance between Mars and Saturn is $1.19666\times {10}^9$ kms. A spaceship starts from Earth and flies to Saturn via Mars.
Find the total distance covered by the spaceship in metres.

1. 1.41919x1012 m
2. 14191900000
3. 1419190000000
4. 1.41919 $\times$ ${10}^9$ km

Q.

The distance between Earth and Mars is 2.2253x${10}^8$ kms. The distance between Mars and Saturn is $1.19666\times {10}^9$ kms. A spaceship starts from Earth and travels to Saturn via Mars. Find the total distance covered by the spaceship in metres.

1. 1.41919x1012 m
2. 14191900000
3. 1419190000000
4. 1.41919x${10}^9$ km

Q.

Question 84

In the following question, state whether the given statements are true (T) or false (F).

The exponential form for
$(-2{)}^4\times {\left(\frac{5}{2}\right)}^{4}is(5^4).$

Q.

The relation $x^m÷x^n=x^{m-n}$ is true, if $x>0$ and $m\le n$.

1. True
2. False

Q. State whether true or false
${\left(2\right)}^5\times {\left(4\right)}^5={\left(24\right)}^5$.

1. True
2. False

Q.

Simplify : $\frac{6^{\frac{2}{3}}\times \sqrt[3]{6^7}}{\sqrt[3]{6^6}}$

Q.

${\left(\frac{3}{4}\right)}^5÷{\left(\frac{5}{3}\right)}^5$ is equal to

1. ${(\frac{3}{4}÷\frac{5}{3})}^0$

2. ${(\frac{3}{4}÷\frac{5}{3})}^{10}$

3. ${(\frac{3}{4}÷\frac{5}{3})}^5$

4. ${(\frac{3}{4}÷\frac{5}{3})}^1$