**Exponents with decimal and fractional bases :**

Here we are going to see some examples problems on exponents with decimal and fractional bases.

**How to evaluate exponents with decimal bases ?**

- To evaluate an expression with decimal bases, first we have to convert the decimal number as integer.
- For that we have to multiply it by 10, 100, 1000,etc.
- Once we convert the decimal as fraction, we can distribute the power for both numerator and denominator separately.

**How to evaluate exponents with fractional bases ?**

**To evaluate exponents with fractional bases, we have to distribute the power for numerator and denominator separately.**

**Let us see some examples based on the above concepts.**

**Example 1 :**

**Solve the following expressions**

**(2/5)**³

**Solution :**

Since the given question is a fraction, we have to distribute the powers.

= 2³/3³

2³ means we have to multiply the base (2) three times like wise 3³ means we have to multiply 3 three times.

= (2 x 2 x 2)/(3 x 3 x 3)

= 8/27

Hence the answer is 8/27.

**Example 2 :**

**Solve the following expressions**

**(0.2)**³

**Solution :**

Since the given question is in the form of decimal, we have to convert this into a fraction, then we can easily evaluate the exponent.

In order to remove the decimal point, we have to multiply and divide both numerator and denominator by 10.

0.2 = 0.2 x (10/10) = 2/10

= **(2/10)**³

= **2**³**/10**³

= (2 x 2 x 2)/(10 x 10 x 10)

= 8/1000

Since we have three zeroes in the denominator, we have to move the decimal point three digits to the left.

Hence the answer is 0.008.

**Example 3 :**

**Solve the following expressions**

**(0.29)**²

**Solution :**

Since the given question is in the form of decimal, we have to convert this into a fraction, then we can easily evaluate the exponent.

In order to remove the decimal point, we have to multiply and divide both numerator and denominator by 100.

0.29 = 0.29 x (100/100) = 29/100

= **(29/100)**²

= (29 x 29)**/(100 x 100)**

= 841/10000

Since we have four zeroes in the denominator, we have to move the decimal point four digits to the left.

Hence the answer is 0.0841.

**Example 4 :**

**Solve the following expressions**

**(0.01)**³

**Solution :**

Since the given question is in the form of decimal, we have to convert this into a fraction, then we can easily evaluate the exponent.

In order to remove the decimal point, we have to multiply and divide both numerator and denominator by 100.

0.01 = 0.01 x (100/100) = 1/100

= **(1/100)**³

= (1 x 1 x 1)**/(100 x 100 x 100)**

= 1/1000000

Since we have four zeroes in the denominator, we have to move the decimal point six digits to the left.

Hence the answer is 0.000001

**Example 5 :**

**Solve the following expressions**

**(5/2)**³

**Solution :**

Since the given question is a fraction, we have to distribute the powers.

= 5³/2³

= (5 x 5 x 5)/(2 x 2 x 2)

= 125/8

Hence the answer is 125/8.

After having gone through the stuff given above, we hope that the students would have understood "Exponents with decimal and fractional bases".

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