Simplify and express each of the following in exponential form:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

(i) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

=

=

=

= (a^m × aⁿ = a^{m + n})

Using identity: (a^m aⁿ = a^{m - n})

= 2^{5 - 5} × 3^{4 – 1}

= 2⁰3³

= 1 × 3³

= 3³

(ii) In the above question,

We have to simplify the given numbers into exponential form:

[(5²)³ × 5⁴] 5⁷

Using identity: (a^m)ⁿ = a^mn)

= [(5)^{2 × 3} × 5⁴] 5⁷

= [(5)⁶ × 4] 5⁷

Using identity: (a^m × aⁿ = a^{m + n})

= [5^{6 + 4}] 5⁷

Using identity: (a^m aⁿ = a^{m - n})

Therefore,

= 5¹⁰ 5⁷

= 5^{10 – 7}

= 5³

(iii) In the above question,

We have to simplify the given numbers into exponential form:

We have,

25⁴ 5³

= (5 × 5)⁴ 5³

Using identity: (a^m)ⁿ = a^mn

= 5^{2 × 4} 5³

= 5⁸ 5³

Using identity: (a^m aⁿ = a^{m - n})

5⁸ 5³

= 5^{8 – 3}

= 5⁵

(iv) In the above question,

We have to simplify the given numbers into exponential form:

We have,

=

Using identity: (a^m aⁿ = a^{m - n})

= 3^{1 - 1} × 7^{2 – 1} × 11^{8 – 3}

= 3⁰ × 7¹ × 11⁵

= 1 × 7 × 11⁵

= 7 × 11⁵

(v) In the above question,

We have to simplify the given numbers into exponential form:

We have,

Using identity: (a^m × aⁿ = a^{m + n})

=

Using identity (a^m aⁿ = a^{m - n})

= 3^{7 - 7}

= 3^o

= 1

(vi) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

2⁰ + 3⁰ + 4⁰

= 1 + 1 + 1

= 3

(vii) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

2⁰ × 3⁰ × 4⁰

= 1 × 1 × 1

= 1

(viii) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

(3⁰ + 2⁰) × 5⁰

= (1 + 1) × 1

= 2

(ix) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

=

Using identity: (a^m)ⁿ = a^mn

Using identity: (a^m aⁿ = a^{m - n})

= 2^{8 – 6} × a^{5 - 3}

= 2² × a²

Using identity [a^m ×b^m = (a × b)^m]

= (2 × a)²

= (2a)²

(x) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

() × a⁸

Using identity: (a^m aⁿ = a^{m - n})

= a^{5 – 3} × a⁸

= a² × a⁸

Using identity (a^m × aⁿ = a^{m + n})

= a^{2 + 8}

= a¹⁰

(xi) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

Using identity: (a^m aⁿ = a^{m - n})

= 4^{5 – 5} × a^{8 – 5} × b^{3 – 2}

= 4⁰ × a³ × b¹

= 1 × a³ × b

= a³b

(xii) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

(2³ × 2)²

Using identity: (a^m × aⁿ = a^{m + n})

= (2^{3 – 1})²

= (2⁴)²

Using identity: (a^m)ⁿ = a^mn

Therefore,

= 2^{4 × 2}

= 2⁸