Informative line

Advanced Operations On Variables

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 \(y\).

Illustration Questions

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: \(a×a×a×a×a×a×a\)

Illustration Questions

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

\(b^2c^3\)

Illustration Questions

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:

\(x×x×x×y×y×z\)

Illustration Questions

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.  

Illustration Questions

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 \(4a^2 bc^3\)

Illustration Questions

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  \(7× a × b × c× c× c\) 

Illustration Questions

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

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