**Exponents with integer bases : **

An integer is a number that can be written without a fractional component.

Before going to see example problems in this topic, first we have to see what is exponent and what is base.

In this topic we will have four kind of questions,

- Positive base with positive exponent
- Positive base with negative exponent
- Negative base with positive exponent
- Negative base with negative exponent

The exponents of a number says, how many times to use the number in a multiplication.

For example, if we have 5^{2} we have to multiply the base that is 5 two times.

5^{2 }= 5 x 5 = 25

Let us see some example problems based on the above concept.

**Example 1 :**

Evaluate 10⁷

**Solution :**

Base = 10

Exponent = 7

To evaluate this we have to multiply the base(10) seven times.

That is,

10^{7} = 10 x 10 x 10 x 10 x 10 x 10 x 10

= 10000000

Hence the value of 10⁷ is 10000000.

**Example 2 :**

Evaluate 8²

**Solution :**

Base = 8

Exponent = 2

To evaluate this we have to multiply the base(8) two times.

That is,

8² = 8 x 8

= 64

Hence the value of 8² is 64.

Whenever we have negative exponent for the positive base, first we have to make the power as positive, for that we have to write the reciprocal of base and change the negative power as positive.

For example,

5⁻² = (1/5)² = (1/5) x (1/5) = 1/25

Let us see some example problem based on the above concept.

**Example 3 :**

Evaluate 4⁻⁴

**Solution :**

Base = 4

Exponent = -4

To evaluate this first we have to make the power as positive

That is,

4⁻⁴ = (1/4)⁴

Multiply 1/4 four times

= (1/4) x (1/4) x (1/4) x (1/4) = 1/256

Hence the value of 4⁻⁴ is 1/256.

**Example 4 :**

Evaluate 3⁻²

**Solution :**

Base = 3

Exponent = -2

To evaluate this first we have to make the power as positive

That is,

3⁻² = (1/3)²

Multiply 1/3 two times

= (1/3) x (1/3) = 1/9

Hence the value of 3⁻² is 1/9.

The exponents of a number says how many times to use the number in a multiplication.

For example, if we have (-5)² we have to multiply the base that is -5 two times.

(-5)² = (-5) x (-5) = 25

Instead of repeating negative sign, we can follow a simple method to decide whether the answer will have positive sign or negative sign.

- If we have odd number as power then the answer will have negative sign.
- If we have even number as power then the answer will have positive sign.

Let us see some example problem based on the above concept.

**Example 5 :**

Evaluate (-7)⁴

**Solution :**

Base = -7

Exponent = 4

To evaluate this we have to multiply the base(-7) four times.

That is,

(-7)⁴ = (-7) x (-7) x (-7) x (-7)

= 2401

Since we have even power, the answer will have positive sign. It is enough to multiply 7 four times. Don't have to repeat negative sign.

Hence the value of (-7)⁴ is 2401

**Example 6 :**

Evaluate (-1)²

**Solution :**

Base = -1

Exponent = 2

To evaluate this we have to multiply the base(-1) two times.

That is,

(-1)² = (-1) x (-1)

= 1

Since we have even power, the answer will have positive sign. It is enough to multiply 1 two times. Don't have to repeat negative sign.

Hence the value of (-1)² is 1.

Whenever we have negative base and negative exponent, first we have to convert the negative exponent as positive. For that we have to take the reciprocal of base.

So, the power the will become positive. Now we have to check whether the power is odd or even.

- If it is odd the answer will have negative sign.
- If it is even the answer will have positive sign.

Let us see some example problem based on the above concept.

**Example 7 :**

Evaluate (-7)⁻⁴

**Solution :**

Base = -7

Exponent = -4

(-7)⁻⁴ = (-1/7)⁴

Since the power is even the answer will have positive sign. Now we have to multiply 1/7 four times.

= (1/7) x (1/7) x (1/7) x (1/7) = 1/2401

Hence the value of (-7)⁻⁴ is 1/2401.

**Example 8 :**

Evaluate (-6)⁻³

**Solution :**

Base = -6

Exponent = -3

(-6)⁻³ = (-1/6)³

Since the power is odd, the answer will have negative sign. Now we have to multiply 1/6 two times.

= (1/6) x (1/6) x (1/6) = -1/216

Hence the value of (-1/6)³ is -1/216.

After having gone through the stuff given above, we hope that the students would have understood how to solve problems in exponents with integer bases.

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