CBSE Class 8 Mental Maths Exponents And Powers Worksheet - Free Printable

The image you provided is a summary of key points and properties related to Exponents and Powers. Below, I will explain each point in detail:

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Points to Remember

1. Exponent is used to express very large numbers and very small numbers in standard form.
- Exponents allow us to write very large or very small numbers in a more manageable form using powers of 10.
- For example:
- Large number: \( 1,230,000,000 \) can be written as \( 1.23 \times 10^9 \).
- Small number: \( 0.000000005 \) can be written as \( 5 \times 10^{-9} \).

2. Very large numbers can be expressed in standard form using positive exponents.
- Example: The distance between the Sun and the Moon is approximately \( 1.49 \times 10^{11} \) meters.
- Here, \( 10^{11} \) indicates that the decimal point is moved 11 places to the right.

3. Very small numbers can be expressed in standard form using negative exponents.
- Example: 1 micron (which is \( 0.000001 \) meters) can be written as \( 10^{-6} \) meters.
- Here, \( 10^{-6} \) indicates that the decimal point is moved 6 places to the left.

4. If \( x, y, a, b \) are rational numbers, then the following properties hold:

- (i) \( x^a \times x^b = x^{a+b} \)
- When multiplying two powers with the same base, add the exponents.
- Example: \( 2^3 \times 2^4 = 2^{3+4} = 2^7 \).

- (ii) \( x^a \div x^b = x^{a-b} \)
- When dividing two powers with the same base, subtract the exponents.
- Example: \( 5^6 \div 5^2 = 5^{6-2} = 5^4 \).

- (iii) \( x^a \times y^a = (xy)^a \)
- When multiplying two different bases raised to the same exponent, multiply the bases and keep the exponent.
- Example: \( 3^2 \times 4^2 = (3 \times 4)^2 = 12^2 \).

- (iv) \( x^a \div y^a = \left( \frac{x}{y} \right)^a \)
- When dividing two different bases raised to the same exponent, divide the bases and keep the exponent.
- Example: \( 8^3 \div 2^3 = \left( \frac{8}{2} \right)^3 = 4^3 \).

- (v) \( \left( \frac{x}{y} \right)^a = \frac{x^a}{y^a} \)
- Raising a fraction to a power means raising both the numerator and the denominator to that power.
- Example: \( \left( \frac{2}{3} \right)^2 = \frac{2^2}{3^2} = \frac{4}{9} \).

- (vi) \( (x^a)^b = x^{ab} \)
- When raising a power to another power, multiply the exponents.
- Example: \( (2^3)^2 = 2^{3 \times 2} = 2^6 \).

- (vii) \( x^a \times y^{-a} = \frac{x^a}{y^a} = \left( \frac{x}{y} \right)^a \)
- Multiplying a base by another base raised to a negative exponent is equivalent to dividing the first base by the second base, both raised to the same exponent.
- Example: \( 5^3 \times 2^{-3} = \frac{5^3}{2^3} = \left( \frac{5}{2} \right)^3 \).

- (viii) \( x^0 = 1 \)
- Any non-zero number raised to the power of 0 is equal to 1.
- Example: \( 7^0 = 1 \), \( (-3)^0 = 1 \).

- (ix) \( x^{-a} = \frac{1}{x^a} \)
- A negative exponent indicates the reciprocal of the base raised to the positive exponent.
- Example: \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \).

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Explanation of Each Property

1. Addition of Exponents (\( x^a \times x^b = x^{a+b} \)):
- This property arises because exponents represent repeated multiplication. For example:
\[
x^3 \times x^2 = (x \cdot x \cdot x) \cdot (x \cdot x) = x^{3+2} = x^5
\]

2. Subtraction of Exponents (\( x^a \div x^b = x^{a-b} \)):
- Division of powers with the same base involves canceling out common factors. For example:
\[
\frac{x^5}{x^2} = \frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x} = x^{5-2} = x^3
\]

3. Product of Bases (\( x^a \times y^a = (xy)^a \)):
- This property follows from the distributive nature of exponents over multiplication. For example:
\[
2^3 \times 3^3 = (2 \cdot 3)^3 = 6^3
\]

4. Quotient of Bases (\( x^a \div y^a = \left( \frac{x}{y} \right)^a \)):
- Similar to the product of bases, but for division. For example:
\[
\frac{4^2}{2^2} = \left( \frac{4}{2} \right)^2 = 2^2
\]

5. Power of a Fraction (\( \left( \frac{x}{y} \right)^a = \frac{x^a}{y^a} \)):
- Raising a fraction to a power means applying the exponent to both the numerator and the denominator. For example:
\[
\left( \frac{3}{4} \right)^2 = \frac{3^2}{4^2} = \frac{9}{16}
\]

6. Power of a Power (\( (x^a)^b = x^{ab} \)):
- When raising a power to another power, the exponents are multiplied. For example:
\[
(2^3)^2 = 2^{3 \times 2} = 2^6
\]

7. Negative Exponent (\( x^{-a} = \frac{1}{x^a} \)):
- A negative exponent signifies the reciprocal of the base raised to the positive exponent. For example:
\[
5^{-2} = \frac{1}{5^2} = \frac{1}{25}
\]

8. Zero Exponent (\( x^0 = 1 \)):
- Any non-zero number raised to the power of 0 is defined to be 1. This is consistent with the division rule of exponents:
\[
\frac{x^a}{x^a} = x^{a-a} = x^0 = 1
\]

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Conclusion

These properties are fundamental to working with exponents and powers. They simplify calculations involving very large or very small numbers and provide a systematic way to manipulate expressions with exponents.

Thus, the solution to the problem of understanding these points is to memorize and apply these rules consistently.

\boxed{\text{Properties of Exponents and Powers explained}}