Question

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Answer 1

Question

Simplify:

$\dfrac{a^{2n+3} \times a^{(2n + 1)(n+2)} }{(a^{3})^{2n+1} \times a^{n(2n+1)}}$

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Answer

First, we'll simplify the exponential part of the given expression by monomial and binomial multiplication.

$\implies \dfrac{a^{2n+3} \times a^{(2n + 1)(n+2)} }{(a^{3})^{2n+1} \times a^{n(2n+1)}}$

$\implies \dfrac{a^{2n+3} \times a^{2n(n+2)+1(n+2)} }{(a^{3})^{2n+1} \times a^{2n^{2}+n}}$

$\implies \dfrac{a^{2n+3} \times a^{2n^{2}+4n+n+2} }{(a^{3})^{2n+1} \times a^{2n^{2}+n}}$

$\implies \dfrac{a^{2n+3} \times a^{2n^{2}+5n+2} }{(a^{3})^{2n+1} \times a^{2n^{2}+n}}$

Apply this Law of Exponent -: $(a^{m})^{n} = a^{m\times n}$

$\implies \dfrac{a^{2n+3} \times a^{2n^{2}+5n+2} }{(a^{3})^{2n+1} \times a^{2n^{2}+n}}$

$\implies \dfrac{a^{2n+3} \times a^{2n^{2}+5n+2} }{a^{3(2n+1)} \times a^{2n^{2}+n}}$

$\implies \dfrac{a^{2n+3} \times a^{2n^{2}+5n+2} }{a^{6n+3} \times a^{2n^{2}+n}}$

Apply this Law of Exponent -: $a^{m} + a^{n} = a^{m+n}$

$\implies \dfrac{a^{2n+3} \times a^{2n^{2}+5n+2} }{a^{6n+3} \times a^{2n^{2}+n}}$

$\implies \dfrac{a^{2n+3+2n^{2}+5n+2} }{a^{6n+3+2n^{2}+n}}$

$\implies \dfrac{a^{7n+5+2n^{2}} }{a^{7n+3+2n^{2}}}$

Apply this Law of Exponent -: $\dfrac{a^{m}}{a^{n}} = a^{m-n}$

$\implies \dfrac{a^{7n+5+2n^{2}} }{a^{7n+3+2n^{2}}}$

$\implies {a^{(7n+5+2n^{2}) - (7n + 3 + 2n^{2})} }$

$\implies {a^{7n+5+2n^{2} - 7n - 3 - 2n^{2}} }$

$\implies {a^{7n- 7n +5- 3 +2n^{2} - 2n^{2}} }$

$\implies a^{2}$

$\therefore \bf \dfrac{a^{2n+3} \times a^{(2n + 1)(n+2)} }{(a^{3})^{2n+1} \times a^{n(2n+1)}} = a^{2}$

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Answer 2

Question:

$\sf{\dfrac{{a}^{2n + 3} \times {a} ^{(2n + 1)(n + 2)}}{({a}^{3})^{2n + 1} \times {a}^{n(2n + 1)}}}$

Solution:

Here, we have been asked to simplify the given equation.

— Let us firstly simplify the denominator of the equation by using indices rules. The law we firstly need to use is:

$\leadsto \sf{{({x}^{m})^{n}={x}^{m\times n}}}$

$\longrightarrow\: \: \: \sf{\dfrac{{a}^{2n + 3} \times {a} ^{(2n + 1)(n + 2)}}{{a}^{3\times (2n + 1)} \times {a}^{2{n}^{2}+1n}}}$

Performing multiplication on denominator's exponents.

$\longrightarrow\: \: \: \sf{\dfrac{{a}^{2n + 3} \times {a} ^{(2n + 1)(n + 2)}}{{a}^{6n+3} \times {a}^{2{n}^{2}+n}}}$

Now, we can observe that the denominator is in the form of $\sf{{x}^{m}\times {x}^{n}}$.

We know,

$\leadsto\sf{{x}^{m}\times {x}^{n}={x}^{m+n}}$

Using, this law in the equation.

$\longrightarrow\: \: \: \sf{\dfrac{{a}^{2n + 3} \times {a} ^{(2n + 1)(n + 2)}}{{a}^{6n+3+2{n}^{2}+n}}}$

Performing addition on the denominator's exponent.

$\longrightarrow\: \: \: \sf{\dfrac{{a}^{2n + 3} \times {a} ^{(2n + 1)(n + 2)}}{{a}^{7n+3+2{n}^{2}}}}$

Now, let's simplify the numerator part.

We observe that the numerator of the equation is in the form of $\sf{{x}^{m}\times {x}^{n}}$. We know,

$\leadsto\sf{{x}^{n}\times {x}^{n}={x}^{m+n}}$

Using this law in the equation,

$\longrightarrow\: \: \: \sf{\dfrac{{a}^{2n+3+(2n+1)(n+2)}}{{a}^{7n+3+2{n}^{2}}}}$

$\longrightarrow\: \: \: \sf{\dfrac{{a}^{2n+3+2{n}^{2}+5n+2}}{{a}^{7n+3+2{n}^{2}}}}$

Performing addition on numerator's exponents.

$\longrightarrow\: \: \: \sf{\dfrac{{a}^{2{n}^{2}+7n+5}}{{a}^{7n+3+2{n}^{2}}}}$

Now, we observe that the whole equation is in the form of $\sf{\dfrac{{x}^{m}}{{x}^{n}}}$

We know,

$\leadsto\sf{\dfrac{{x}^{m}}{{x}^{n}}={x}^{m-n}}$

Using the law in the equation, we get:

$\longrightarrow\: \: \: \sf{{a}^{(2{n}^{2}+7n+5)-(2{n}^{2}-7n-3)}}$

$\longrightarrow\: \: \: \sf{{a}^{2{n}^{2}+7n+5-2{n}^{2}-7n-3}}$

Cancelling +2n² with -2n² and +7n with -7n.

$\longrightarrow\: \: \: \sf{{a}^{5-3}}$

Performing subtraction.

$\longrightarrow\: \: \: \sf{{a}^{2}}$

Required Answer:

Therefore, $\bold{\dfrac{{a}^{2n + 3} \times {a} ^{(2n + 1)(n + 2)}}{({a}^{3})^{2n + 1} \times {a}^{n(2n + 1)}}={a}^{2}}$