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Exponential Rules And Properties

Zero and One Rules of Power

  • We need to learn about the properties of powers to evaluate any expression.
  • Here, we will learn the zero & one rules of power.

One rules of power

  • Rule 1: One to any power is always \(1\).

For example: 

(i) \(1^3\) can be written as \(1^3=1×1×1=1\)

(ii) \(1^5\) can be written as \(1^5=1×1×1×1×1=1\)

  • Rule 2: Any number to the power \(1\) is always that number.

For example:

\(\text{(i)}\;2^1=2\\\text{(ii)}\,3^1=3\\\text{(iii)}(100)^1=100\)

Zero rules of power

  • Rule 1: Any number to the power zero is always \(1\).

For example:

\(\text{(i)}\;2^0=1\\\text{(ii)}\;5^0=1\\\text{(iii)}\;6^0=1\\\text{(iv)}\;(10)^0=1\)

  • Rules 2: Zero to any power is zero.

For example:

\(\text{(i)}\;0^5=0\\\text{(ii)}\;0^{10}=0\\\text{(iii)}\;0^{12}=0\)

Illustration Questions

What is the equivalent term of \(5^0\,?\)

A \(1\)

B \(0\)

C \(5\)

D \(5×0\)

×

We know that any number to the power zero is always \(1\).

Thus, \(5^0=1\)

The equivalent term of \(5^0\) is \(1\).

Hence, option (A) is correct.

What is the equivalent term of \(5^0\,?\)

A

\(1\)

.

B

\(0\)

C

\(5\)

D

\(5×0\)

Option A is Correct

Negative Exponents

  • Any number to the negative power is equal to the reciprocal of the number to the same positive power.

e.g.  \(a^{-x}=\dfrac{1}{a^x}\)

For example:

\(\text{(i)}\;6^{-2}=\dfrac{1}{6^2}\\\text{(ii)}\;7^{-3}=\dfrac{1}{7^3}\\\text{(iii)}\;5^{-6}=\dfrac{1}{5^6}\)

Illustration Questions

Which one of the following options is equivalent to \(5^{-2}\,?\)

A \(-5^2\)

B \(\dfrac{1}{5^2}\)

C \(\dfrac{1}{5^{-2}}\)

D \(5×(-2)\)

×

We know that any number to the negative power is equal to the reciprocal of the number to the same positive power.

\(a^{-x}=\dfrac{1}{a^x}\)

Thus, \(5^{-2}=\dfrac{1}{5^2}\)

Hence, option (B) is correct.

Which one of the following options is equivalent to \(5^{-2}\,?\)

A

\(-5^2\)

.

B

\(\dfrac{1}{5^2}\)

C

\(\dfrac{1}{5^{-2}}\)

D

\(5×(-2)\)

Option B is Correct

Product Rule

  • The product of same numbers with different powers is equal to the number to the sum of powers.

i.e., \(a^x\,a^y=a^{x+y}\)

For example:

\(\text{(i)}\;4^2.\,4^3=4^{2+3}=4^5\)   [Dot \((\cdot)\) implies multiplication]

\(\text{(ii)}\;3^6.\,3^1=3^{6+1}=3^7\)

Illustration Questions

Which one of the following options is equivalent to \(6^3.\,6^5\,?\)

A \(6×3×5\)

B \(6(3+5)\)

C \(6^8\)

D \(8^6\)

×

We know that the product of same numbers with different powers is equal to the number to the sum of powers.

\(a^x\,a^y=a^{x+y}\)

Thus, \(6^3.\,6^5=6^{3+5}=6^8\)

Hence, option (C) is correct.

Which one of the following options is equivalent to \(6^3.\,6^5\,?\)

A

\(6×3×5\)

.

B

\(6(3+5)\)

C

\(6^8\)

D

\(8^6\)

Option C is Correct

Quotient Rule

  • The division of the same numbers with different powers is equal to the number to the difference of powers.

i.e. \(\dfrac{a^x}{a^y}=a^{x-y}\)

For example: 

\(\text{(i)}\;\dfrac{5^3}{5^2}=5^{3-2}=5^1\\ \text{(ii)}\;\dfrac{4^5}{4^2}=4^{5-2}=4^3\\ \text{(iii)}\dfrac{(4)^8}{(4)^4}=4^{8-4}=4^4\)

Illustration Questions

Which one of the following options is equivalent to \(\dfrac{6^9}{6^7}\,?\)

A \(6×2\)

B \(6×(9-7)\)

C \(2^6\)

D \(6^2\)

×

We know that the division of the same numbers with different powers is equal to the number to the difference of powers.

\(\dfrac{a^x}{a^y}=a^{x-y}\)

Thus, \(\dfrac{6^9}{6^7}=6^{9-7}=6^2\)

Hence, option (D) is correct.

Which one of the following options is equivalent to \(\dfrac{6^9}{6^7}\,?\)

A

\(6×2\)

.

B

\(6×(9-7)\)

C

\(2^6\)

D

\(6^2\)

Option D is Correct

Rule of Power of Product

  • The product of different numbers to a power is equal to the product of numbers separately to the same power.

i.e., \((x.y.z)^a=x^a.y^a.z^a\)

For example:

\(\text{(i)}\;(3×4)^3=3^3×4^3\)

\(\text{(ii)}\;(2×6)^5=2^5×6^5\)

Illustration Questions

Which one of the following options is equivalent to \((5×2)^5\)?

A \(5×5×2×5\)

B \((5+2)^5\)

C \(5^5×2^5\)

D \((5)^{5×2}\)

×

We know that the product of different numbers to a power is equal to the product of numbers separately to the same power.

\((x.y.z)^a=x^a.y^a.z^a\)

Thus, \((5×2)^5=5^5×2^5\)

Hence, option (C) is correct.

Which one of the following options is equivalent to \((5×2)^5\)?

A

\(5×5×2×5\)

.

B

\((5+2)^5\)

C

\(5^5×2^5\)

D

\((5)^{5×2}\)

Option C is Correct

Rule of Power of Quotient

  • The division of numbers to a power is equal to the division of numbers separately to the same power.

i.e., \(\left(\dfrac{x}{y}\right)^a=\dfrac{x^a}{y^a}\)

For Example:

\(\text{(i)}\;\left(\dfrac{5}{3}\right)^6=\dfrac{5^6}{3^6}\)

\(\text{(ii)}\;\left(\dfrac{3}{2}\right)^2=\dfrac{3^2}{2^2}\)

\(\text{(iii)}\;\left(\dfrac{6}{7}\right)^3=\dfrac{6^3}{7^3}\)

Illustration Questions

Which one of the following options is equivalent to \(\left(\dfrac{2}{8}\right)^5\)?

A \((2-8)^5\)

B \(\dfrac{2^5}{8^5}\)

C \(5×\dfrac{2}{8}\)

D \(5+\dfrac{2}{8}\)

×

We know that the division of numbers to a power is equal to the division of numbers separately to the same power.

\(\left(\dfrac{x}{y}\right)^a=\dfrac{x^a}{y^a}\)

Thus, \(\left(\dfrac{2}{8}\right)^5=\dfrac{2^5}{8^5}\)

Hence, option (B) is correct.

Which one of the following options is equivalent to \(\left(\dfrac{2}{8}\right)^5\)?

A

\((2-8)^5\)

.

B

\(\dfrac{2^5}{8^5}\)

C

\(5×\dfrac{2}{8}\)

D

\(5+\dfrac{2}{8}\)

Option B is Correct

Power Rule

  • The number with a power raised to another power is equal to the number to the product of powers.

i.e., \((a^m)^n=a^{m×n}\)

For example:

\(\text{(i)}\;(5^2)^3=5^{2×3}=5^6\)

\(\text{(ii)}\;(6^2)^5=6^{2×5}=6^{10}\)

\(\text{(iii)}\;(7^3)^6=7^{3×6}=7^{18}\)

Note: \((a^m)^n\neq a^{m^n}\)

Example:

\((7^2)^3=7^{2×3}=7^{6}\)

and

\((7^{2^3})=7^{8}\)

Illustration Questions

Which one of the following options is equivalent to \((3^6)^5\)?

A \((3×6)^5\)

B \(3^{30}\)

C \((3×5)^6\)

D \(3×6×5\)

×

We know that the number with a power raised to another power is equal to the number to the product of powers.

\((a^m)^n=a^{m×n}\)

Thus, \((3^6)^5=3^{6×5}=3^{30}\)

Hence, option (B) is correct.

Which one of the following options is equivalent to \((3^6)^5\)?

A

\((3×6)^5\)

.

B

\(3^{30}\)

C

\((3×5)^6\)

D

\(3×6×5\)

Option B is Correct

Writing a Number in Exponential form

  • A number can be expressed in the exponential form by evaluating it.
  • For this, first, prime factorize the number.
  • Then, write the prime factors in the exponential form.

Example: \(32\)

Prime factors of \(32=2×2×2×2×2\)

Exponential form of \(32=2^5\)

Illustration Questions

Which term is equivalent to \(2^5.4^3.8\)?

A \(2^7\)

B \(2^9\)

C \(2^{10}\)

D \(2^{14}\)

×

\(2^5.4^3.8\)

We first simplify this expression

\(=2^5.(2.2)^3.(2.2.2)\)

\(=2^5.2^3.2^3.2^3\)     [By product rule]

\(=2^{5+3+3+3}\)

\(=2^{14}\)

Hence, option (D) is correct.

Which term is equivalent to \(2^5.4^3.8\)?

A

\(2^7\)

.

B

\(2^9\)

C

\(2^{10}\)

D

\(2^{14}\)

Option D is Correct

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