CONVERSION BETWEEN PRODUCT FORM AND EXPONENTIAL FORM

Problem 1 :

Which power of 9 is equal to 3⁸ ?

Solution :

First let us find the expansion of 3⁸,

= 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3

Let us try to get 9 from here. Combining 3 x 3, we will get 9.

= 9 x 9 x 9 x 9

= 9⁴

So, 9⁴ is equal to 3⁸.

Problem 2 :

The value of (-2)⁵ is 

a) -32     b)  10    c) -1/32    d) 1/32

Solution :

= (-2)⁵

Since we have negative sign at the base and we have odd exponent, the answer should also have negative sign.

= - (2⁵)

= - (2 x 2 x 2 x 2 x 2)

= -32

So, option c is correct.

Problem 3 :

.Which of the following is not true?

a) 2⁴ = 4²     b) 2⁴ = 8    c) 2³ < 3²

Solution :

Option a :

2⁴ = 4² 

So, option a is true.

Option b :

2⁴ = 8

Writing 8 in exponential form, we get

8 = 2 x 2 x 2

= 2³

So, option b is not true.

Problem 4 :

Write the base and exponent of (-1/3)⁴

Solution :

= (-1/3)⁴

Since we have negative sign at the base and we have even number at the power, we will have positive sign for the result.

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

= 1/81

Problem 5 :

Write 7.2 x10³ in usual form.

Solution :

= 7.2 x 10³

By writing 10³ in expanded form, we get

= 7.2 x 10 x 10 x 10

= 7.2 x 1000

= 7200

Problem 6 :

Which power of 7 is 343?

Solution :

343 = 7 x 7x 7

= 7³

So, the required power is 3.

Problem 7 :

Express (9^-4)³ as a single exponent of 3?

Solution :

= (9^-4)³

When we have power raised by another power, we have to multiply the powers.

= (9^-12)

To change the negative exponent to positive exponent, we have to write the base as its reciprocal.

= 1/9¹²

Writing 9 in exponential form, we get

= 1/(3²)¹²

= 1/3²⁴

Problem 8 :

Express -8/125 as a power of rational number.

Solution :

= -8/125

8 = 2 x 2 x 2

= 2³

125 = 5 x 5 x 5

= 5³

= -(2³/ 5³)

= (-2/5)³

Problem 9 :

Express the following in the form of p/q.

(-3/7)³ ÷ (6/7)²

Solution :

= (-3/7)³ ÷ (6/7)²

= (-3/7)³ x (7/2)²

= (-3)³ / (7)³ x (7)² / (2)²

= -27/7(4)

= -27/28

Problem 10 :

Write the reciprocal of (-6/7)⁵ ÷ 7⁰

Solution :

= (-6/7)⁵ ÷ 7⁰

Anything to the power is 1.

= (-6/7)⁵ ÷ 1

= (-6/7)⁵

Reciprocal is (-7/6)⁵

Problem 11 :

Express (-11)² x (-7)² in single exponential form.

Solution :

= (-11)² x (-7)²

Since two terms are having same exponents and they are multiplying, we can use the property.

a^m x b^m = (a x b)^m

= (-11 x (-7))²

= 77²

Problem 12 :

Find the value of x if (-3)^{x - 2} = -243

Solution :

(-3)^{x - 2} = -243

Expressing -243 in exponential form, we get

(-3)^{x - 2} = (-3)⁵

Since we have expressed both terms with the same base and we can equate the bases then power will also be equal.

x - 2 = 5

x = 5 + 2

x = 7

So, the value of x is 7.

Problem 12 :

Which is greater 3¹² or 6⁶

Solution :

3¹² or 6⁶

Comparison can be done simply by changing the base or power as the same.

3¹² = (3²)⁶

= 9⁶

Now comparing 9⁶ and 6⁶, we know that  9⁶ is greater. that is 3¹²

Problem 13 :

Which power of 8 is equal to 2⁶

a) 3     b) 2    c)  1     d)  4

Solution :

Let us expand 2⁶ to find the value.

2⁶ = 2x 2 x 2 x 2 x 2

= 64

8² = 64

So, the answer is option b.

Problem 14 :

Which of the following is greater?

a)  4^-2     b)  4^-3    c) 3^-4    d) 3^-2

Solution :

Evaluating each option, we can do the comparison.

Here numerators are the same, then comparing the denominator. By dividing 1 by smaller value, we will get larger number as result. Then option d is correct.

Problem 15 :

Which power of 9 is equal to 3⁸ ?

Solution :

3⁸ can be written as (3²)⁴

Evaluating the base 3², we get 9.

(3²)⁴ = 9⁴

So, the answer is 4.

Problem 16 :

Write the exponential form for 9 x 9 x 9 taking base as 3

Solution :

= 9 x 9 x 9

= (3 x 3) x (3 x 3) x (3 x 3)

= 3⁶

Problem 17 :

Compare 3.8 x 10²⁷ and 1.9 x 10²⁸

Solution :

Trying to express the base or power as the same.

= 3.8 x 10²⁷ and 1.9 x 10²⁸

= (2 x 1.9 x 10²⁷) 

= 3.8 x 10²⁷ -----(1)

1.9 x 10²⁸ = 1.9 x 10²⁷ x 10

= 19 x 10²⁷ ------(2)

By comparing (1) and (2), we get (2) is greater. That is,

1.9 x 10²⁸

Problem 18 :

If

3³⁰⁰⁰ – 3²⁹⁹⁹- 3²⁹⁹⁸ + 3²⁹⁹⁷ = a.3²⁹⁹⁸

 then find the value of a.

Solution :

3³⁰⁰⁰ – 3²⁹⁹⁹- 3²⁹⁹⁸ + 3²⁹⁹⁷ = a.3²⁹⁹⁸

3²⁹⁹⁷(3³ - 3² - 3 + 1) = a.3²⁹⁹⁸

(27 - 9 - 3 + 1) = a(3)

16 = 3a

a = 16/3

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.