Informative line

# 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:

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

#### 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:

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

#### 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]

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

#### 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:

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

#### 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:

$$\text{(i)}\;(3×4)^3=3^3×4^3$$

$$\text{(ii)}\;(2×6)^5=2^5×6^5$$

#### 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:

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

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

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

#### 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

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

#### 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 $$32=2^5$$

#### 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