Question

please tell me fast ​

Answer 1

Given

$\sf\to \bigg(\dfrac{a^{-1}b^2}{a^2b^{-4}} \bigg)^7\div \bigg(\dfrac{a^{3}b^{-5}}{a^{-2}b^{3}} \bigg)^7=a^xb^y$

To find

$\sf\to x+y$

Now take

$\sf\to \bigg(\dfrac{a^{-1}b^2}{a^2b^{-4}} \bigg)^7\div \bigg(\dfrac{a^{3}b^{-5}}{a^{-2}b^{3}} \bigg)^7$

$\sf\to\bigg\{\bigg(\dfrac{a^{-1}}{a^2} \bigg)\times\bigg(\dfrac{b^2}{b^{-4}} \bigg)\bigg\}^7\div\bigg\{\bigg(\dfrac{a^{3}}{a^{-2}} \bigg)\times\bigg(\dfrac{b^{-5}}{b^{3}} \bigg)\bigg\}^7$

$\sf\to\bigg\{\bigg(\dfrac{a^{-1}}{a^2} \bigg)\times\bigg(\dfrac{b^2}{b^{-4}} \bigg)\bigg\}^7\times\bigg\{\bigg(\dfrac{a^{-2}}{a^{3}} \bigg)\times\bigg(\dfrac{b^{3}}{b^{-5}} \bigg)\bigg\}^7$

Use Exponential Law

$\sf\to\bigg(\dfrac{a^m}{a^n} \bigg)=a^{m-n}$

$\sf\to\{a^m\}^n=a^{mn}$

$\sf\to a^m\times a^n=a^{m+n}$

We get

$\sf\to \{(a^{-1-2})\times(b^{2-(-4)})\}^7\times\{(a^{-2-3})\times (b^{3-(-5)})\}^7$

$\sf\to \{(a^{-3})\times(b^{6})\}^7\times\{(a^{-5})\times (b^{8})\}^7$

$\sf\to \{(a^{-3\times7})\times(b^{6\times7})\}\times\{(a^{-5\times7})\times (b^{8\times7})\}$

$\sf\to \{(a^{-21})\times(b^{42})\}\times\{(a^{-35})\times (b^{56})\}$

$\sf\to a^{-21}\times b^{42}\times a^{-35}\times b^{56}$

$\sf\to a^{-21}\times a^{-35}\times b^{56}\times b^{42}$

$\sf\to a^{-21-35}\times b^{56+52}$

$\sf\to a^{-56}\times b^{108}$

Now compare with

$\sf\to a^x \times b^y$

We get

$\sf\to x= -56 \:\:and \: y = 108$

now we have to find x + y , we get

$\sf\to -56+108$

$\sf\to 52$

Answer

$\sf\to 52$

Answer 2

Required answer:-

Question:

$◍If \: ( \frac{a {}^{ - 1}b {}^{2} }{a {}^{2}b {}^{ - 4} } ) {}^{7} \div ( \frac{a {}^{3}b {}^{ -5} }{a {}^{ - 2}b {}^{3} } ) {}^{ 7} = a {}^{x} .b {}^{y} \: find \: x + y$

Solution:

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To find:

◍ Value of x + y

Concept used:

First understand the concept that is of Indices(Exponents).

If m is a positive integer, then a × a × a × a-----upto m terms, is written as a^m; where 'a' is called the base and 'm' or 'a raised to the power ( or exponent or index ).

a^m is read as 'a power m' or 'a raised to the ' power m'.

Laws of Indices (Exponents) used:

◍ a^(m)× a^(n) = a^(m+n)

◍(a^m)^n = a^(mn)

◍(a^m / a^n ) = a^(m-n)

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Kindly refer the attachment for answer and further explanation due to latex problem.

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