Question

1. Find the value of n, when
(5/3)^2n + 1 (5/3)^5 = (5/3^)n + 2 .

Answer 1

The value of n is -3.

Given:

$(\frac{5}{3}) ^{2n}+1(\frac{5}{3})^5 = (\frac{5}{3})^{n+2}$

To Find:

The value of n.

Solution:

According to the law of exponents, on multiplying numbers having different powers with the same base, their powers get added up, i.e. if a, b, m, and n are integers, then:

1. $a^{m}$ x $a^{n} = a^{(m+n)}$

2.  $(\frac{a}{b})^{m}$ x $(\frac{a}{b})^{n} = (\frac{a}{b})^{(m+n)}$

We have been given that,

$(\frac{5}{3}) ^{2n}+1(\frac{5}{3})^5 = (\frac{5}{3})^{n+2}$

⇒ $(\frac{5}{3}) ^{2n}+(\frac{5}{3})^5 = (\frac{5}{3})^{n+2}$

Adding the powers in the LHS as they have the same base, we get:

$(\frac{5}{3}) ^{2n+5}= (\frac{5}{3})^{n+2}$

Since the base of both sides is the same, we can compare their powers to get the value of n. Hence, we get

2n+5 = n+2

⇒ n = -3.

Hence the value of n is -3.

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Answer 2

Final solution: The value of n = -3.

Step-by-step explanation:

STEP-1:

As per the expression given in the above question.

Given ,

${ (\frac{5}{3}) }^{2n} + {( \frac{5}{3} )}^{5} = {( \frac{5}{3} )}^{n + 2}$

To find - Value of n

STEP-2:

According to the law of exponents we have different expression to solve the different values .

When the same base having different powers , their powers get added up, i.e. if a, b, m, and n are integers, then:

${x}^{a} + {x}^{b} = {x}^{(a + b)}$

Similarly , when the base in fraction ,decimal we use the same expression like above ,but power will be integer.

STEP-3:

HERE,

${ (\frac{5}{3}) }^{2n} + {( \frac{5}{3} )}^{5} = {( \frac{5}{3} )}^{n + 2}$

Add the powers according to law ,

${ (\frac{5}{3}) }^{2n + 5} = {( \frac{5}{3} )}^{n + 2}$

When the two function in equal then , power will be equal

$2n + 5 = n + 2$

Shift the value 5 toward Right hand side ,

$2n = n + 2 - 5$

$2n - n = - 3$

$n = - 3$

Hence ,

The value of n is -3.

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