### Lesson No. 2:

**Solved Problems in Exponents:**

**Example 1:**

**Simplify: **

**a. (81)**^{3/4}

**Solution:**

(81)^{3/4} = (3^{4})^{3/4} = (3)^{4 × 3/4} = (3)^{4 × 3/4} = 3^{3} = 27

**b.(64/25)**^{-3/2}

**Solution:**

(64/25)^{-3/2} = (8^{2}/5^{2})^{-3/2}

=(8/5)^{2}-3/2 = (8/5)^{2 ×-3/2} = (8/5)^{2 ×-3/2}

(8/5)^{-3} = (5/8)^{3} = 5^{3}/8^{3} = 125/512

**c. If 10**^{p} = 0.000001, then find p.

Solution:

0.000001 = 1/1000000 = 1/10^{6} = 10^{-6}

Now, 10^{p} = 10^{-6}

So we get p = -6

**Example 2:**

**Express 2**^{3} + 2^{4} as a product of two prime factors.

Solution:

First of all note that:

2^{3} + 2^{4} ‡ 2^{7}

Now, in 2^{3} and 2^{4} , 2^{3} is a common factor

2^{3} + 2^{4} = 2^{3} (1+2) = 2^{3} × 3

**Example 3: **

**Find the greatest prime factor of:**

2^{3} + 2^{4} +2^{5} + 2^{6} + 2^{7}

Solution:

2^{3} + 2^{4} +2^{5} + 2^{6} + 2^{7} = 2^{3} (1 + 2 + 2^{2} + 2^{3} + 2^{4}) = 2^{3} × 31

Since 31 is a prime number, and the sum

2^{3} + 2^{4} +2^{5} + 2^{6} + 2^{7} is same as 2^{3} × 31.

Therefore, 31 is the greatest prime factor in

2^{3} + 2^{4} +2^{5} + 2^{6} + 2^{7}

**Example 4: **

**Simplify**

2^{6} × 2^{7} - 2^{4} × 2^{9}

Solution:

2^{6} × 2^{7} = 2^{6 + 7} = 2^{13} and 2^{4} × 2^{9} = 2^{4 + 9} = 2^{13}

Therefore, 2^{6} × 2^{7} - 2^{4} × 2^{9} = 2^{13} - 2^{13} = 0

**Example 5:**

**Find 2**^{3} + 2^{-3} + 2^{0}

Solution:

We know that 2^{-3} = 1/2^{3} = 1/8 and 2^{0} = 1

So, 2^{3} + 2^{-3} + 2^{0} = 8 + 1/8 + 1 = 9 + 1/8

LCM is 8, and therefore 9 + 1/8 = (9 × 8 + 1)/8 = (72 + 1)/8 = 73/8

**Example 6:**

**Let a and b be positive integers such that **

a^{b} = 121, then find b^{a} .

Solution:

Since a and b are positive integers and

a^{b} = 121, so, we can write 11^{2} =121

Therefore, a = 11 and b = 2.

Now, b^{a} = 2^{11} = 2048

**Example 7: **

**Compare**

a. -2^{3} and -2^{4}.

b. -1/2^{3 }and -1/2^{4}

Solution:

-2^{3} = -8 and -2^{4 }= -16

Now, -16 < -8

Therefore, -2^{3} > -2^{4}

Very Important Tip:

(-2)^{4} = -2 × -2 × -2 × -2 = 16, but -2^{4} = -16

The negative sign of the base will be gone, only when the base is inside a parentheses and the exponent is an even integer.

b. -1/2^{3} = -1/8, and -1/2^{4} = -1/16

Now, we know that:

8 < 16, -8 > -16 and again -1/8 < -1/16

Very Important Tip:

The inequality sign reverses direction in two cases:

Case 1:

When negative sign is either introduced or pulled out.

For Example:

2 < 3, but -2 > -3 Or

-10 > -11, but 10 < 11

Case 2:

When reciprocals are taken or removed:

The reciprocal of any number x, except 0 is 1/x

The reciprocal of 3 is 1/3

**Example: **

10 < 14 but 1/10 > 1/14 Or

1/2 > 1/3, but 2 < 3, and again

2 < 3, but -2 > -3, and again -1/2 < -1/3

Example 8:

Which of the two is larger?

^{3}√5 or ^{4}√7

Solution:

^{3}√5 is a **surd** of order 4 and ^{4}√7 is a surd of order 3.

**To compare two surds:**

First find LCM of the two orders: i.e. 3 and 4.

LCM of 3 and 4 is 3 × 4 = 12

Now, ^{3}√5 = 5^{1/3} = (5^{4/3×4}) = 5^{4/12} = (625)^{1/12} = 625^{1/12}

Again ^{4}√7 = 7^{1/4} = (7^{3/4×3}) = 7^{3/12} = (7^{3})^{1/12} = 343^{1/12}

Now, 625^{1/12} > 343^{1/12}