EXPONENTS WITH DECIMAL AND FRACTIONAL BASES

About "Exponents with decimal and fractional bases"

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

  =  2³/10³

=  (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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