**Exponents and square roots :**

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

For example, 5³ means we have to multiply the base 5 three times.

That is,

5³ = 5 x 5 x 5 = 125

The square root of a number is the value such that, when a number multiplied by itself,for example

3 x 3 = 9

It is written with a radical symbol " √ " and the number or expression inside the radical symbol is called the radicand.

**Example 1 :**

Simplify the following

(-4)⁵ ÷ (-4)⁸

**Solution :**

**(-4)⁵ ÷ (-4)⁸ = **(-4)⁵ / (-4)⁸

Since both terms are having same base, we have to put only one base and subtract the powers.

= (-4)⁵⁻⁸ ==> (-4) ⁻³

= 1/(-4)³ ==> -1/64

**Example 2 :**

Simplify the following

3⁴ x 3⁻³

**Solution :**

3⁴ x 3⁻³

If we want to multiply two or more terms which are multiplying with the same base, then we have to put only one base and add the powers.

= 3⁴⁻³ ==> 3

**Example 3 :**

Simplify the following

(3/8)⁵ x (3/8)⁴ ÷ (3/8)⁹

**Solution :**

= (3/8)^(5 + 4 - 9)

= (3/8)^(9 - 9)

= (3/8)^⁰ ==> 1

**Example 4 :**

Simplify the following

(2²)³/(3²)²

**Solution :**

= (2²)³/(3²)²

= 2⁶/3⁴ ==> 32/81

**Example 5 :**

Find the value of x

(2/8)^(2x) (2/8)^x = (2/8)^6

**Solution :**

(2/8)^(2x) (2/8)^x = (2/8)^6

Put only one base and add the powers,

(2/8)^(2x+x) = (2/8)^6

(2/8)^(3x) = (2/8)^6

Since we have same bases on both sides, the powers are also equal.

3x = 6

Dividing by 3 on both sides, we get x = 2

= 2⁶/3⁴ ==> 32/81

**Example 6 :**

Find the square root of the following

5 x 5 x 11 x 11 x 7 x 7

**Solution :**

= √(5 x 5 x 11 x 11 x 7 x 7)

Inside the radical sign, if the same number is repeated twice, take one number out of the radical sign.

So we can take one 5, one 11 and one 7.

= 5 x 11 x 7 ==> 385

**Example 7 :**

Find the square root of the following

√(25a²) x √(49a)

**Solution :**

= √(25a²) x √(49a)

= √(5 x 5 x a x a x 7 x 7 x a)

= 5 x a x 7 √a ==> 245 a

**Example 8 :**

Find the square root of the following

∛125g³

**Solution :**

= ∛125g³

= ∛(5 x 5 x 5 x g x g x g)

Inside the cube root, if the same number is repeated three times, take one number out of the radical sign.

= 5 g

**Example 9 :**

Find the square root of the following

√(9/64)

**Solution :**

Take square root for both numerator and denominator separately,

= √9/√64

= √(3x3)/√(8x8)

= 3/8

**Example 10 :**

Find the square root of the following

√2.56

**Solution :**

In order to remove the decimal point we have to multiply and divide by 100.

= √2.56 x (100/100)

= √(256/100)

= √(16 x 16)/(10x10) ==> 16/10 ==> 1.6

= √(3x3)/√(8x8)

= 3/8

After having gone through the stuff given above, we hope that the students would have understood "Exponents and square roots".

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