EXPONENTS USING REPEATED MULTIPLICATION

Power of a number can be evaluated using repeated multiplication of the number by itself.

For example, 5 to the third power can be evaluated using multiplication of 5 by itself for three times as shown below.

5³ = 5 x 5 x 5

The power or exponent of a number says how many times a number has to be multiplied repeatedly by itself.

Here, 5 is the base and 3 is the power of exponent.

Repeated Multiplication with Negative Base

(-3)⁴ = -3 x -3 x -3 x -3 = 81

(-3)⁵ = -3 x -3 x -3 x -3 x -3 = -243

The result of (-3)⁴ is positive and the result of (-3)⁵ is negative.

When a negative number is multiplied repeatedly by itself for even number of times, the result is positive and the same is done for odd number of times, the result is negative.

That is, if a negative number has even exponent, its result is positive and when it has odd exponent, its result is negative.

-3² = -(3 x 3) = -9

Even though the exponent of -3² is even, the result is still negative. Because the exponent is applicable only for the number 3 and not for the negative sign.

Solved Problems

Problem 1 :

Write the following expression in exponential form.

m x m x n x n

Solution :

m x m x n x n = m²n²

Problem 2 :

Find the value of 7² x 5².

Solution :

7² x 5² = 7 x 7 x 5 x 5

= 49 x 25

= 1225

Hence, the value of 7 x 7 x 5 x 5 is 1225.

Problem 3 :

Find the value of 5 x 5 x 5.

Solution :

5 x 5 x 5 = 125

Problem 4 :

Write the following expression in exponential form.

13 x b x b x b x b

Solution :

13 x b x b x b x b = 13b⁴

Problem 5 :

Write the following expression in exponential form.

17 x 17 x w x w x w

Solution :

17 x 17 x w x w x w = 17²w²

Problem 6 :

Write the following expression in simplified form.

5 x 5 x p x p x p

Solution :

5 x 5 x p x p x p = 25p³

Problem 7 :

Evaluate (-5)³

Solution :

(-5)³ = -5 x -5 x -5

= -125

Problem 8 :

Evaluate (-5)⁴.

Solution :

(-5)⁴ = -5 x -5 x -5 x -5

= 625

Problem 9 :

Evaluate -5⁴.

Solution :

-5⁴ = - 5 x 5 x 5 x 5

= -625

Problem 10 :

Evaluate 7^-3.

Solution :

7^-3 = 1/7³

= 1/(7 x 7 x 7)

= 1/343

Evaluate the following.

Problem 11 :

5 − 2²

Solution :

= 5 − 2²

We have to multiply 2 two times.

= 5 - 4

= 1

Problem 12 :

7 + 3²

Solution :

= 7 + 3²

We have to multiply 3 two times.

= 7 + 9

= 16

Problem 13 :

9² + 1

Solution :

= 9² + 1

We have to multiply 9 two times.

9² = 9 x 9

= 81 + 1

= 82

Problem 14 :

6² − 5²

Solution :

= 6² − 5²

We have to multiply 6 two times and 5 two times.

6² = 6 x 6

5² = 5 x 5

= 36 - 25

= 11

Problem 15 :

(7 − 2)²

Solution :

= (7 − 2)²

First we have to subtract and then perform for square.

= 5²

5² = 5 x 5

= 25

Problem 16 :

(4 + 3)²

Solution :

= (4 + 3)²

First we have to add and then perform for square.

= 7²

= 49

Problem 17 :

(1 + 2)³

Solution :

= (1 + 2)³

First we have to add and then perform for cube

= 3³

3³ = 3 x 3 x 3

= 27

Problem 18 :

(2 + 8)³

Solution :

= (2 + 8)³

First we have to add and then perform for cube

= 10³

10³ = 10 x 10 x 10

= 1000

Problem 19 :

10 − √16

Solution :

= 10 − √16

= 10 − √4 x 4

= 10 - 4

= 6

Problem 20 :

√(2 + 14)

Solution :

= √(2 + 14)

= √16

= √4 x 4

= 4

Problem 21 :

√4 + 3²

Solution :

= √4 + 3²

First we have to take square root of 4 and then calculate the square.

= √2 x 2 + 9

= 2 + 9

= 11

Problem 22 :

2 x 5 − √4

Solution :

= 2 x 5 − √4

= 2 x 5 − √(2 x 2)

= 10 - 2

= 8

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