Question

write formulas for solving roots problem

Answer 1

In this maths tutorial, we introduce exponents / powers and roots using formulas, solved examples and practice questions.

Powers and Roots | Formulas, Solved Examples, Practice Problems

Exponents, also called powers, are a way of expressing a number multiplied by itself by a certain number of times.

When we write a number a, it is actually a1, said as a to the power 1.

a2 = a*a

a3 = a*a*a
:
:
an = a*a*a*a* . . . n times.

Basic formulas in Powers and Roots

Some basic formulas used to solve questions on exponents are:

(am)n = (an)m = amnam.an = am+na-m = 1/amam/an = am-n = 1/an-m(ab)n = anbn(a/b)n = an/bna0 = 1

 

22 = 4. 23 = 8. This is what we learn in exponents.

√4 = 2. 3√8 = 2. This is what we learn in roots.

Here, √ is called the square root or of 2nd order.

3√ is called the cubeth root or of 3rd order.

Similarly we can have the root of a number of any order.

n√a is called a surd of order n.

The symbol n√ is called radical sign,

n is called the order of the surd and

a is called the radicand.

Some basic formulae used to solve questions on roots are:

n√a = a1/nn√ab = n√a* n√bn√(a/b) = n√a / n√b(n√a)n = a

 

Solved examples in Powers & Roots

Let us consider some examples:

Problem 1. Simplify (7.5*105) / (25*10-4)

Solution:

(7.5*105) / (25*10-4)

→ (75*104) / (25*10-4)

Cancelling 75 with 3 times 25 and applying the formula of am/an = am-n

→ 3*104-(-4)

→ 3*108

 

Problem 2. Find x if 32x-1 + 32x+1= 270.

Solution:

Taking out a term common, we get

→ 32x-1 (1+32)

Observe that here, we applied the formula am+n = am.an in writing 32x+1 as a product of 32x-1 and 32.

→ 32x-1 (10) = 270

→ 32x-1 = 27

→ 32x-1 = 33

→ 2x-1 = 3

→ x = 2.

 

Problem 3. Simplify [10 [ (216)1/3+ (64)1/3 ]3 ] 3/4

Solution:

[10 [ (63)1/3 + (43)1/3 ]3 ] 3/4

→ [ 10 [6 + 4]3 ]3/4

→ [ 10 (10)3 ]3/4

→ (104)3/4

→ 103 = 1000.

 

Problem 4. Simplify [40.08 * (20.22)2 ]10 / [160.16 * (24)0.74 * (42)0.1]

Solution:

[40.08 * (20.22)2]10 / [160.16 * (24)0.74 * (42)0.1]

Applying the formula (am)n = (an)m to the underlined part,

→ [40.08 * (22)0.22]10 / [160.16 * (24)0.74 * (42)0.1]

→ [40.08 * 40.22]10 / [160.16 * (24)0.74 * (42)0.1]

Applying the formula am.an = am+n to the numerator,

→ [40.08+0.22]10 / [160.16 * (24)0.74* (42)0.1]

Simplifying the denominator,

→ [40.3]10 / [(42)0.16 * (42)0.74 * (42)0.1]

Applying the formula am.an = am+n

→ 43 / [(42)0.16+0.74+0.1]

→ 43 / (42)1

Applying the formula am/an = am-n,

= 4.