Class 8 Mathematics worksheet focusing on exponents and powers, including simplification, standard form, and exponential expressions.

Worksheet: Exponents and Powers

Here is the step-by-step solution to each problem in the worksheet:

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#### 1. Simplify: \((4^{-1} - 2^{-1})^3\)

- First, simplify the terms inside the parentheses:
\[
4^{-1} = \frac{1}{4}, \quad 2^{-1} = \frac{1}{2}
\]
So,
\[
4^{-1} - 2^{-1} = \frac{1}{4} - \frac{1}{2} = \frac{1}{4} - \frac{2}{4} = -\frac{1}{4}
\]

- Now, raise the result to the power of 3:
\[
\left(4^{-1} - 2^{-1}\right)^3 = \left(-\frac{1}{4}\right)^3 = -\frac{1}{64}
\]

Answer:
\[
\boxed{-\frac{1}{64}}
\]

---

#### 2. Find the value of:
a) \(\left(\frac{-8}{15}\right)^0 + \left(\frac{16}{5}\right)^0 \times \left(\frac{4}{5}\right)^{-1}\)

- Any number raised to the power of 0 is 1:
\[
\left(\frac{-8}{15}\right)^0 = 1, \quad \left(\frac{16}{5}\right)^0 = 1
\]

- Simplify \(\left(\frac{4}{5}\right)^{-1}\):
\[
\left(\frac{4}{5}\right)^{-1} = \frac{5}{4}
\]

- Substitute these values into the expression:
\[
\left(\frac{-8}{15}\right)^0 + \left(\frac{16}{5}\right)^0 \times \left(\frac{4}{5}\right)^{-1} = 1 + 1 \times \frac{5}{4} = 1 + \frac{5}{4} = \frac{4}{4} + \frac{5}{4} = \frac{9}{4}
\]

b) \(\left[\left(\frac{-2}{3}\right)^2\right]^{-1}\)

- First, simplify \(\left(\frac{-2}{3}\right)^2\):
\[
\left(\frac{-2}{3}\right)^2 = \frac{(-2)^2}{3^2} = \frac{4}{9}
\]

- Now, take the reciprocal (raise to the power of -1):
\[
\left[\left(\frac{-2}{3}\right)^2\right]^{-1} = \left(\frac{4}{9}\right)^{-1} = \frac{9}{4}
\]

Answers:
a) \(\boxed{\frac{9}{4}}\)
b) \(\boxed{\frac{9}{4}}\)

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#### 3. Multiplicative inverse of \(14^4\) is

- The multiplicative inverse of a number \(x\) is \(\frac{1}{x}\). For \(14^4\), the multiplicative inverse is:
\[
\frac{1}{14^4}
\]

Answer:
\[
\boxed{\frac{1}{14^4}}
\]

---

#### 4. The value of \(5^3 \times 3^{-1}\) is

- Simplify \(5^3\):
\[
5^3 = 125
\]

- Simplify \(3^{-1}\):
\[
3^{-1} = \frac{1}{3}
\]

- Multiply the results:
\[
5^3 \times 3^{-1} = 125 \times \frac{1}{3} = \frac{125}{3}
\]

Answer:
\[
\boxed{\frac{125}{3}}
\]

---

#### 5. Simplify and write the answer in exponential form with positive exponents.

a) \((-4)^4 \times (-4)^{-8}\)

- Use the property of exponents \(a^m \times a^n = a^{m+n}\):
\[
(-4)^4 \times (-4)^{-8} = (-4)^{4 + (-8)} = (-4)^{-4}
\]

- Convert to positive exponent:
\[
(-4)^{-4} = \frac{1}{(-4)^4} = \frac{1}{256}
\]

b) \(2^8 \div 2^3\)

- Use the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):
\[
2^8 \div 2^3 = 2^{8-3} = 2^5
\]

c) \(\left(\frac{2}{3}\right)^{-1} \times \left(\frac{2}{3}\right)^{-2} \times \left(\frac{2}{3}\right)^{-3}\)

- Use the property of exponents \(a^m \times a^n \times a^p = a^{m+n+p}\):
\[
\left(\frac{2}{3}\right)^{-1} \times \left(\frac{2}{3}\right)^{-2} \times \left(\frac{2}{3}\right)^{-3} = \left(\frac{2}{3}\right)^{-1 + (-2) + (-3)} = \left(\frac{2}{3}\right)^{-6}
\]

- Convert to positive exponent:
\[
\left(\frac{2}{3}\right)^{-6} = \left(\frac{3}{2}\right)^6
\]

d) \((-2)^7 \div (-2)^4\)

- Use the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\):
\[
(-2)^7 \div (-2)^4 = (-2)^{7-4} = (-2)^3 = -8
\]

Answers:
a) \(\boxed{\frac{1}{256}}\)
b) \(\boxed{2^5}\)
c) \(\boxed{\left(\frac{3}{2}\right)^6}\)
d) \(\boxed{-8}\)

---

#### 6. Simplify:

a) \(\frac{4^3 \times a^5 \times b^4}{4^2 \times a^3 \times b^5}\)

- Simplify the powers of 4:
\[
\frac{4^3}{4^2} = 4^{3-2} = 4^1 = 4
\]

- Simplify the powers of \(a\):
\[
\frac{a^5}{a^3} = a^{5-3} = a^2
\]

- Simplify the powers of \(b\):
\[
\frac{b^4}{b^5} = b^{4-5} = b^{-1} = \frac{1}{b}
\]

- Combine the results:
\[
\frac{4^3 \times a^5 \times b^4}{4^2 \times a^3 \times b^5} = 4 \times a^2 \times \frac{1}{b} = \frac{4a^2}{b}
\]

b) \(\frac{10^{-4} \times 9^{-3}}{2^{-3} \times 15^{-4}}\)

- Rewrite the expression using positive exponents:
\[
\frac{10^{-4} \times 9^{-3}}{2^{-3} \times 15^{-4}} = \frac{\frac{1}{10^4} \times \frac{1}{9^3}}{\frac{1}{2^3} \times \frac{1}{15^4}} = \frac{2^3 \times 15^4}{10^4 \times 9^3}
\]

- Simplify the numerator and denominator:
\[
15^4 = (3 \times 5)^4 = 3^4 \times 5^4, \quad 10^4 = (2 \times 5)^4 = 2^4 \times 5^4, \quad 9^3 = (3^2)^3 = 3^6
\]

So,
\[
\frac{2^3 \times 15^4}{10^4 \times 9^3} = \frac{2^3 \times 3^4 \times 5^4}{2^4 \times 5^4 \times 3^6} = \frac{2^3 \times 3^4}{2^4 \times 3^6} = \frac{1}{2 \times 3^2} = \frac{1}{18}
\]

Answers:
a) \(\boxed{\frac{4a^2}{b}}\)
b) \(\boxed{\frac{1}{18}}\)

---

#### 7. Find \(x'\) if:
a) \(4^{x+1} = 16^{x-3}\)

- Express 16 as a power of 4:
\[
16 = 4^2 \implies 16^{x-3} = (4^2)^{x-3} = 4^{2(x-3)}
\]

- Equate the exponents:
\[
4^{x+1} = 4^{2(x-3)} \implies x + 1 = 2(x - 3)
\]

- Solve for \(x\):
\[
x + 1 = 2x - 6 \implies 1 + 6 = 2x - x \implies x = 7
\]

b) \(2^{x+3} = 1\)

- Any number raised to the power of 0 is 1:
\[
2^{x+3} = 1 \implies 2^{x+3} = 2^0 \implies x + 3 = 0 \implies x = -3
\]

Answers:
a) \(\boxed{7}\)
b) \(\boxed{-3}\)

---

#### 8. Find \(n'\) if:
a) \(\left(\frac{-2}{5}\right)^{-2} - \left(\frac{-2}{5}\right)^{-3} = \frac{4}{25}\)

- Simplify \(\left(\frac{-2}{5}\right)^{-2}\):
\[
\left(\frac{-2}{5}\right)^{-2} = \left(\frac{5}{-2}\right)^2 = \frac{25}{4}
\]

- Simplify \(\left(\frac{-2}{5}\right)^{-3}\):
\[
\left(\frac{-2}{5}\right)^{-3} = \left(\frac{5}{-2}\right)^3 = \frac{125}{-8} = -\frac{125}{8}
\]

- Substitute these values into the equation:
\[
\frac{25}{4} - \left(-\frac{125}{8}\right) = \frac{25}{4} + \frac{125}{8} = \frac{50}{8} + \frac{125}{8} = \frac{175}{8}
\]

- This does not match \(\frac{4}{25}\), so there might be a misunderstanding in the problem statement. Let's assume the problem is correctly stated and solve for \(n\) in a general sense.

b) \((25)^n = 5^n\)

- Express 25 as a power of 5:
\[
25 = 5^2 \implies (25)^n = (5^2)^n = 5^{2n}
\]

- Equate the exponents:
\[
5^{2n} = 5^n \implies 2n = n \implies n = 0
\]

Answers:
a) (Assuming correct setup, no specific \(n\) matches directly.)
b) \(\boxed{0}\)

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#### 9. Write in standard form:

a) \(836000000\)

- Move the decimal point 8 places to the left:
\[
836000000 = 8.36 \times 10^8
\]

b) \(0.00000045\)

- Move the decimal point 7 places to the right:
\[
0.00000045 = 4.5 \times 10^{-7}
\]

c) \(0.00000306\)

- Move the decimal point 6 places to the right:
\[
0.00000306 = 3.06 \times 10^{-6}
\]

d) \(\frac{6103}{1000}\)

- Simplify the fraction:
\[
\frac{6103}{1000} = 6.103
\]

Answers:
a) \(\boxed{8.36 \times 10^8}\)
b) \(\boxed{4.5 \times 10^{-7}}\)
c) \(\boxed{3.06 \times 10^{-6}}\)
d) \(\boxed{6.103}\)

---

#### 10. Write in usual form:

a) \(6.34 \times 10^{-5}\)

- Move the decimal point 5 places to the left:
\[
6.34 \times 10^{-5} = 0.0000634
\]

b) \(8.9 \times 10^4\)

- Move the decimal point 4 places to the right:
\[
8.9 \times 10^4 = 89000
\]

c) \(6 \times 10^4\)

- Move the decimal point 4 places to the right:
\[
6 \times 10^4 = 60000
\]

d) \(2.3456 \times 10^3\)

- Move the decimal point 3 places to the right:
\[
2.3456 \times 10^3 = 2345.6
\]

Answers:
a) \(\boxed{0.0000634}\)
b) \(\boxed{89000}\)
c) \(\boxed{60000}\)
d) \(\boxed{2345.6}\)

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Final Answer:

\[
\boxed{
\begin{array}{l}
1. -\frac{1}{64} \\
2. \text{a) } \frac{9}{4}, \text{ b) } \frac{9}{4} \\
3. \frac{1}{14^4} \\
4. \frac{125}{3} \\
5. \text{a) } \frac{1}{256}, \text{ b) } 2^5, \text{ c) } \left(\frac{3}{2}\right)^6, \text{ d) } -8 \\
6. \text{a) } \frac{4a^2}{b}, \text{ b) } \frac{1}{18} \\
7. \text{a) } 7, \text{ b) } -3 \\
8. \text{a) } \text{(No specific } n \text{ matches)}, \text{ b) } 0 \\
9. \text{a) } 8.36 \times 10^8, \text{ b) } 4.5 \times 10^{-7}, \text{ c) } 3.06 \times 10^{-6}, \text{ d) } 6.103 \\
10. \text{a) } 0.0000634, \text{ b) } 89000, \text{ c) } 60000, \text{ d) } 2345.6
\end{array}
}
\]