Informative line

# Power of Variables

## Power

• The power of a variable denotes the value up to which the variable is multiplied by itself.
• It is written at the top right of a number.

Example: $$x^2$$

Here, $$2$$ is written at the top right of $$x$$

Thus, the power of $$x$$ is 2.

• The power is also known as an exponent or an index.

## Base

• The base is the variable on which power is raised.

Example: $$z^4$$

Here, the base is $$z$$.

Note: Instead of writing $$y^1$$, we write only $$y$$.

#### What is the power in the expression $$y^7$$?

A $$y$$

B $$7$$

C Both

D None of these

×

Power is written at the top right of a number.

Thus, $$7$$ is the power of $$y$$.

Hence, option (B) is correct.

### What is the power in the expression $$y^7$$?

A

$$y$$

.

B

$$7$$

C

Both

D

None of these

Option B is Correct

# Product Form

• An exponent can be written in product form.
• In product form, we multiply the literal with itself repeatedly up to the value of power.

For example: $$a^7$$

Here, the power is  $$7$$, so ' $$a$$'  is to be multiplied $$7$$ times by itself.

Product form: $$a×a×a×a×a×a×a$$

#### Choose the product form of the given expression:                           $$c^4$$

A $$c× 4$$

B $$c × c × c × c$$

C $$c+c+c+c$$

D $$4^c$$

×

In product form, we multiply a variable repeatedly with itself up to the value of power.

Here, the power of $$c$$ is 4 so, $$c$$ is to be multiplied $$4$$ times with itself.

Thus, the expanded form is

$$c × c × c × c$$

Hence, option (B) is correct.

### Choose the product form of the given expression:                           $$c^4$$

A

$$c× 4$$

.

B

$$c × c × c × c$$

C

$$c+c+c+c$$

D

$$4^c$$

Option B is Correct

# Exponential form when more than one variable is involved

• The expression in which more than one variable is used can also be written in an exponential form.
• In this, we count the number of times a variable is repeatedly multiplied by itself and write it as a power of that literal.

For example: $$b×b×c×c×c$$

Here, $$b$$ is multiplied $$2$$ times by itself and $$c$$ is multiplied $$3$$ times by itself.

Thus, the exponential form is

$$b^2c^3$$

#### Choose the exponential form of the given term: $$a×a×b×c×c×c$$

A $$abc$$

B $$a×2×b×c×3$$

C $$a^2bc^3$$

D None of these

×

Given: $$a ×a× b×c×c×c$$

Count the number of times a variable is repeatedly multiplied by itself and write it as a power of that variable.

Here, $$a$$ is multiplied $$2$$ times by itself, $$b$$ is multiplied once and $$c$$ is multiplied $$3$$ times by itself.

Thus, the exponential form is $$a^2 bc^3$$.

Hence, option (C) is correct.

### Choose the exponential form of the given term: $$a×a×b×c×c×c$$

A

$$abc$$

.

B

$$a×2×b×c×3$$

C

$$a^2bc^3$$

D

None of these

Option C is Correct

# Product Form When More Than One Variable Is Involved

• The expression in which more than one variable is used can also be written in product form.
• For the product form, we multiply a variable repeatedly by itself up to the value of the power of that variable.

For example: $$x^3 y^2z$$

Here, the power of $$x$$ is $$3$$ and the power of $$y$$ is $$2$$.

Thus, $$x$$ should be multiplied by itself up to $$3$$ times and $$y$$ should be multiplied by itself up to $$2$$ times.

Hence, the product form is:

$$x×x×x×y×y×z$$

#### Choose the product form of the given expression: $$\ell m^2n^4$$

A $$\ell mn$$

B $$\ell ×m×2×n×4$$

C $$\ell ×m$$

D $$\ell ×m×m×n×n×n×n$$

×

Given: $$\ell m^2n^4$$

Multiply the variable repeatedly with itself up to the value of a power of that variable.

Here, the power of  $$\ell$$ is  $$1$$, the power of $$m$$ is $$2$$ and the power of $$n$$ is $$4$$.

So, $$\ell$$ should be multiplied once, $$m$$ should be multiplied by itself up to $$2$$ times and $$n$$ should be multiplied by itself up to $$4$$ times.

Thus, the product form is: $$\ell ×m ×m ×n ×n ×n ×n$$

Hence, option (D) is correct.

### Choose the product form of the given expression: $$\ell m^2n^4$$

A

$$\ell mn$$

.

B

$$\ell ×m×2×n×4$$

C

$$\ell ×m$$

D

$$\ell ×m×m×n×n×n×n$$

Option D is Correct

# Exponential Form

• The product form of a variable can be written in exponential form.
• In this, we count the number of times a variable is repeatedly multiplied by itself and write it as a power of that variable.

For example:

$$m× m× m× m× m$$

Here, $$m$$ is multiplied $$5$$ times by itself.

Thus, we write it as $$m^5$$.

• $$n× n$$ is written as $$n^2$$ and is called $$n$$ squared.
• $$n× n× n$$ is written as $$n^3$$ and is called $$n$$ cubed.
• $$n× n× n× n$$ is written as $$n^4$$ and is called $$n$$ raised to the power $$4$$.
• $$n× n× n× n× n$$ is written as $$n^5$$ and is called $$n$$ raised to the power $$5$$ and so on.

#### Choose the exponential form of the given term: $$s×s×s×s×s×s×s×s×s$$

A $$s^9$$

B $$s×9$$

C $$s$$

D $$9$$

×

Count the number of times a variable is repeatedly multiplied by itself and write it as a power of that variable.

Given: $$s×s×s×s×s×s×s×s×s$$

Here, $$s$$ is multiplied $$9$$ times by itself.

Thus, the exponential form is $$s^9$$.

Hence, option (A) is correct.

### Choose the exponential form of the given term: $$s×s×s×s×s×s×s×s×s$$

A

$$s^9$$

.

B

$$s×9$$

C

$$s$$

D

$$9$$

Option A is Correct

# Conversion of Product Form involving Variables and Numbers into Exponential Form

• Numbers are used along with variables.
• The expressions involving the product of numbers and variables can be written in exponential form.
• In this, we count the number of times a variable is multiplied by itself and write it as the power of that variable.

For example: $$4×a×a×b×c×c×c$$

Here, $$a$$ is multiplied $$2$$ times with itself, $$b$$ is multiplied once and $$c$$ is multiplied $$3$$ times with itself.

Thus, the exponential form is $$4a^2 bc^3$$

#### Choose the exponential form of the given expression: $$5×w×w×w×w×\ell ×\ell×b×b×b×b×b$$

A $$5w\ell b$$

B $$w\ell b$$

C $$5 w^4 \ell ^2 b^5$$

D $$w×4×\ell ×2×b×5$$

×

Given: $$5×w×w×w×w×\ell ×\ell×b×b×b×b×b$$

Count the number of times a variable is multiplied by itself and write it as the power of that variable.

Here, $$w$$ is multiplied $$4$$ times with itself, $$\ell$$ is multiplied $$2$$ times with itself and $$b$$ is multiplied $$5$$ times with itself.

Thus, the exponential form is $$5w^4 \ell^2 b^5$$.

Hence, option (C) is correct.

### Choose the exponential form of the given expression: $$5×w×w×w×w×\ell ×\ell×b×b×b×b×b$$

A

$$5w\ell b$$

.

B

$$w\ell b$$

C

$$5 w^4 \ell ^2 b^5$$

D

$$w×4×\ell ×2×b×5$$

Option C is Correct

# Conversion of Exponential Form involving Variables and Numbers into Product Form

• Numbers are used along with variables.
• The expressions involving numbers and variables having powers can be written in product form.
• For the product form, we multiply a variable by itself up to the value of the power of that variable.

For example: $$7 abc^3$$

Here, the power of $$c$$ is $$3$$.

Thus, in product form, $$c$$ should be multiplied $$3$$ times by itself.

Hence, the product form is  $$7× a × b × c× c× c$$

#### Choose the product form of the given expression: $$2x^2yz^3$$

A $$2× x× 2× y× 3× z$$

B $$x^2 yz^3$$

C $$2× xyz$$

D $$2× x× x× y× z× z×z$$

×

Given: $$2x^2yz^3$$

Multiply a variable with itself up to the value of the power of that variable.

Here, the power of $$x$$ is $$2$$ and $$z$$ is $$3$$.

Thus, $$x$$ should be multiplied by itself up to $$2$$ times and $$z$$ should be multiplied by itself up to $$3$$ times.

So, the product form is

$$2 × x×x×y×z×z×z$$

Hence, option (D) is correct.

### Choose the product form of the given expression: $$2x^2yz^3$$

A

$$2× x× 2× y× 3× z$$

.

B

$$x^2 yz^3$$

C

$$2× xyz$$

D

$$2× x× x× y× z× z×z$$

Option D is Correct