Question

Find the value of n if (81) ^ 5/n =243

Answer 1

ANSWER:

Given:

• 81^(5/n) = 243

To Find:

• Value of n

Solution:

$\text{We are given that,}\\\\:\longrightarrow81^{\frac{5}{n}}=243\\\\\text{So,}\\\\:\implies81^{\frac{5}{n}}=243\\\\:\implies(3^4)^{\frac{5}{n}}=3^5\\\\:\implies3^{4\times\frac{5}{n}}=3^5\\\\:\implies3^{\frac{20}{n}}=3^5\\\\\text{On comparing the exponents,}\\\\:\implies\dfrac{20}{n}=5\\\\\text{On cross-multiplying,}\\\\:\implies5n=20\\\\:\implies n=\dfrac{20\!\!\!\!/^{\:4}}{5\!\!\!/}\\\\\bf{:\implies n=4}$

Formula Used:

• a^(m^n) = a^(mn)

Learn More:

$\begin{gathered}\boxed{\begin{minipage}{5 cm}\bf{\dag}\:\:\underline{\text{Laws of Exponents :}}\\\\\bigstar\:\:\sf\dfrac{a^m}{a^n} = a^{m - n}\\\\\bigstar\:\:\sf{(a^m)^n = a^{mn}}\\\\\bigstar\:\:\sf(a^m)(a^n) = a^{m + n}\\\\\bigstar\:\:\sf\dfrac{1}{a^n} = a^{-n}\\\\\bigstar\:\:\sf\sqrt[\sf n]{\sf a} = (a)^{\dfrac{1}{n}}\end{minipage}}\end{gathered}$

Answer 2

Find the value of n if (81)^5/n = 243.

It is given that we have to find the value of n in $(81) {}^{ \frac{5}{n} } = 243$, To do so, we will;

$(81) {}^{ \frac{5}{n} } = 243$

• Convert both sides of the the equation to have exponents.

→ $(3⁴) {}^{ \frac{5}{n} } = 3⁵$

→ $3 {}^{4 \times \frac{5}{n} } = 3⁵$

→ $3 {}^{ \frac{20}{n} } = 3⁵$

• When compare the given exponents, we get;

→ $\frac{20}{n} = 5$

→ 20 = 5n

→ $\frac{20}{5} = n$

→ 4 = n or n = 4

Thus the value of ‘n’ will be 4.

More Information :-

$\begin{gathered}\begin{gathered}\boxed{\begin{minipage}{5 cm}\bf{\dag}\:\:\underline{\text{Laws of Exponents :}}\\\\\bigstar\:\:\sf\dfrac{a^m}{a^n} = a^{m - n}\\\\\bigstar\:\:\sf{(a^m)^n = a^{mn}}\\\\\bigstar\:\:\sf(a^m)(a^n) = a^{m + n}\\\\\bigstar\:\:\sf\dfrac{1}{a^n} = a^{-n}\\\\\bigstar\:\:\sf\sqrt[\sf n]{\sf a} = (a)^{\dfrac{1}{n}}\end{minipage}}\end{gathered}\end{gathered}$