myhelper: RD Sharma 2020 solution class 9 chapter 2 Exponents of Real Numbers VSAQS

RD Sharma 2020 solution class 9 chapter 2 Exponents of Real Numbers VSAQS

VSAQS

Page-2.33

Question 1:

Write

{(625)}^{- 1 / 4}

in decimal form.

Answer 1:

We have to write

in decimal form. So,

{(625)}^{\frac{- 1}{4}} = \frac{1}{625^{\frac{1}{4}}} = {(\frac{1}{(5^{4})})}^{\frac{1}{4}}

Hence the decimal form of

Question 2:

State the product law of exponents.

Answer 2:

State the product law of exponents.
If

is any real number and

are positive integers, then

By definition, we have

(

factor)

(

factor)

factors

Thus the exponent "product rule" tells us that, when multiplying two powers that have the same base, we can add the exponents.

Question 3:

State the quotient law of exponents.

Answer 3:

The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. If a is a non-zero real number and m, n are positive integers, then

We shall divide the proof into three parts
(i) when

(ii) when

(iii) when

Case 1
When

We have

\frac{a^{m}}{a^{n}} = \frac{a \times a \times a . . . . to m factors}{a \times a \times a . . . . t o n f a c t o r s} \frac{a^{m}}{a^{n}} = a \times a \times a . . . . t o (m - n) f a c t o r s \frac{a^{m}}{a^{n}} = a^{m - n}

Case 2
When

We get

Cancelling common factors in numerator and denominator we get,

By definition we can write 1 as

Case 3
When

In this case, we have

Hence

, whether

or,

Question 4:

State the power law of exponents.

Answer 4:

The "power rule" tell us that to raise a power to a power, just multiply the exponents.
If a is any real number and m, n are positive integers, then

We have,

factors

Hence,

Page-2.34

Question 5:

If 2⁴ × 4² =16^x, then find the value of x.

Answer 5:

We have to find the value of x provided

So,

By equating the exponents we get

Hence the value of x is .

Question 6:

If 3^x^-1 = 9 and 4^y⁺² = 64, what is the value of $\frac{x}{y}$ ?

Answer 6:

We have to find the value of for

So,

By equating the exponent we get

Let’s take

By equating the exponent we get

By substituting in we get

Hence the value of is

Question 7:

Write the value of $\sqrt[3]{7} \times \sqrt[3]{49} .$

Answer 7:

We have to find the value of . So,

By using law rational exponents we get,

Hence the value of is

Question 8:

Write ${(\frac{1}{9})}^{- 1 / 2} \times (64)^{- 1 / 3}$ as a rational number.

Answer 8:

We have to find the value of . So,

Hence the value of the value of is .

Question 9:

Write the value of $\sqrt[3]{125 \times 27}$ .

Answer 9:

We have to find the value of $\sqrt[3]{125 \times 27}$ . So,

$\sqrt[3]{125 \times 27} = \sqrt[3]{5^{3} \times 3^{3}} = 5 \times 3 = 15$

Hence the value of the value of is .

Question 10:

For any positive real number x, find the value of

${(\frac{x^{a}}{x^{b}})}^{a + b} \times {(\frac{x^{b}}{x^{c}})}^{b + c} \times {(\frac{x^{c}}{x^{a}})}^{c + a}$

Answer 10:

We have to find the value of L =

By using rational exponents, we get

By using rational exponents we get

By definition we can write as 1

Hence the value of expression is .

Question 11:

Write the value of ${\{5 (8^{1 / 3} + 27^{1 / 3})^{3}\}}^{1 / 4} .$

Answer 11:

We have to find the value of. So,

By using rational exponents we get

Hence the simplified value of is

Question 12:

Simplify ${[{\{{(625)}^{- 1 / 2}\}}^{- 1 / 4}]}^{2}$

Answer 12:

We have to simplify. So,

Hence, the value of is

Question 13:

For any positive real number x, write the value of

${\{{(x^{a})}^{b}\}}^{\frac{1}{a b}} {\{{(x^{b})}^{c}\}}^{\frac{1}{b c}} {\{{(x^{c})}^{a}\}}^{\frac{1}{c a}}$