**Exponent 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 "Exponent and square roots".

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

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