Exponential Form-Solved Examples

Exponential form

The exponential form of a number is b^{a}.

Here, 

b = base = The number which is multiplied with itself multiple times.

a = exponent (power) = Number of times the base is multiplied with itself.

b^{a} = exponential form = “b” is multiplied repeatedly “a” times. 

When there are more than 2 unique factors of a number, it is written as the product of two exponential numbers.

Examples

1. 8=2 \times 2 \times 2=2^{3}

Base = 2

Exponent = 3

2. 81=3 \times 3 \times 3 \times 3=3^{4}

Base = 3

Exponent = 4

3. 108=2 \times 2 \times 3 \times 3 \times 3=2^{2} \times 3^{3}

The exponent can be a positive integer, negative integer or a fraction.

1. Positive Exponent (b^{a})
   The positive exponent denotes that the base is multiplied with itself “a” number of times.
   
2. Negative Exponent (b^{-a})
   Negative Exponent denotes that the reciprocal of the base is multiplied “a” number of times.

3. Fractional Exponent (b^{\frac{p}{q}})

Fractional Exponent denotes powers and roots of the base together.

b^{\frac{p}{q}}=\sqrt[q]{b^{p}}=(\sqrt[q]{b})^{p}

Note: b^{\frac{1}{q}}=q^{t h} root of b

Solved Examples

1. Write 3375 in its exponential form.

Solution:

Prime factorisation of 3375:

⇒ 3375 = (3 × 3 × 3) × (5 × 5 × 5)

⇒ 3375=3^{3} \times 5^{3}=15^{3}

2. Find the value of 6^{3}.

Solution:

6^{3}=6 \times 6 \times 6=216

Hence the value of 6^{3} is 216.

3. Find the value of 5^{-2}.

Solution

Negative Exponent denotes that the reciprocal of the base is multiplied “a” number of times.

b^{-a}=\frac{1}{b^{a}}

5^{-2}=\frac{1}{5^{2}}=\frac{1}{25}=0.04

Hence the value of 5^{-2} is equal to 0.04.

4. Find the value of 8^{\frac{2}{3}}.

Solution:

b^{\frac{p}{q}}=\sqrt[q]{b^{p}}=(\sqrt[q]{b})^{p}

8^{\frac{2}{3}}=(\sqrt[3]{8})^{2}=2^{2}=4

Hence the value of 8^{\frac{2}{3}} is equal to 4.