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} × a^{n} = a^{m + n})

Using identity: (a^{m} a^{n} = a^{m - n})

= 2^{5 - 5} × 3^{4 – 1}

= 2^{0}3^{3}

= 1 × 3^{3}

= 3^{3}

(ii) In the above question,

We have to simplify the given numbers into exponential form:

[(5^{2})^{3} × 5^{4}] 5^{7}

Using identity: (a^{m})^{n} = a^{mn})

= [(5)^{2 × 3} × 5^{4}] 5^{7}

= [(5)^{6} × 4] 5^{7}

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

= [5^{6 + 4}] 5^{7}

Using identity: (a^{m} a^{n} = a^{m - n})

Therefore,

= 5^{10} 5^{7}

= 5^{10 – 7}

= 5^{3}

(iii) In the above question,

We have to simplify the given numbers into exponential form:

We have,

25^{4} 5^{3}

= (5 × 5)^{4} 5^{3}

Using identity: (a^{m})^{n} = a^{mn}

= 5^{2 × 4} 5^{3}

= 5^{8} 5^{3}

Using identity: (a^{m} a^{n} = a^{m - n})

5^{8} 5^{3}

= 5^{8 – 3}

= 5^{5}

(iv) In the above question,

We have to simplify the given numbers into exponential form:

We have,

=

Using identity: (a^{m} a^{n} = a^{m - n})

= 3^{1 - 1} × 7^{2 – 1} × 11^{8 – 3}

= 3^{0} × 7^{1} × 11^{5}

= 1 × 7 × 11^{5}

= 7 × 11^{5}

(v) In the above question,

We have to simplify the given numbers into exponential form:

We have,

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

=

Using identity (a^{m} a^{n} = 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,

2^{0} + 3^{0} + 4^{0}

= 1 + 1 + 1

= 3

(vii) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

2^{0} × 3^{0} × 4^{0}

= 1 × 1 × 1

= 1

(viii) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

(3^{0} + 2^{0}) × 5^{0}

= (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} a^{n} = a^{m - n})

= 2^{8 – 6} × a^{5 - 3}

= 2^{2} × a^{2}

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,

() × a^{8}

Using identity: (a^{m} a^{n} = a^{m - n})

= a^{5 – 3} × a^{8}

= a^{2} × a^{8}

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

= a^{2 + 8}

= a^{10}

(xi) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

Using identity: (a^{m} a^{n} = a^{m - n})

= 4^{5 – 5} × a^{8 – 5} × b^{3 – 2}

= 4^{0} × a^{3} × b^{1}

= 1 × a^{3} × b

= a^{3}b

(xii) In the above question,

We have to simplify the given numbers into exponential form:

∴ We have,

(2^{3} × 2)^{2}

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

= (2^{3 – 1})^{2}

= (2^{4})^{2}

Using identity: (a^{m})^{n} = a^{mn}

Therefore,

= 2^{4 × 2}

= 2^{8}

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