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Solve $3^{x+1}$ $\times$ $5^{2x-1}$ $=$ $3375$ and Find the value of $x$

$3^{x+1} \times 5^{2x-1} = 3375$

It is an exponential form equation in the combination of both arithmetic and algebraic form and it can be solved in mathematics by using the rules of exponents to obtain the value of $x$.

$3^x \times 3^1 \times \dfrac{5^{2x}}{5} = 3375$

Use product rule of exponents to express $3^{x+1}$ as two multiplying exponential terms having same base and also use division rule of exponents to express $5^{2x-1}$ as the division of two exponential terms having same base.

$\implies 3^x \times 3 \times \dfrac{5^{2x}}{5} = 3375$

$\implies 3^x \times 5^{2x} \times \dfrac{3}{5} = 3375$

$\implies 3^x \times 5^{2x} = 3375 \times \dfrac{5}{3}$

$\implies 3^x \times 5^{2x} = 5 \times \dfrac{3375}{3}$

$\implies \require{cancel} 3^x \times 5^{2x} = 5 \times \dfrac{\cancel{3375}}{\cancel{3}}$

$\implies 3^x \times 5^{2x} = 5 \times 1125$

$\implies 3^x \times 5^{2x} = 5625$

Use power rule of exponents to express the term $5^{2x}$.

$\implies 3^x \times 25^x = 5625$

$\implies 75^x = 5625$

The square of the $75$ is $5625$. So, express the number $5625$ in terms of $75$.

$\implies 75^x = 75 \times 75$

$\implies 75^x = 75^2$

Compare the both exponential terms. The bases are equal. So, the exponents should also be equal.

$\therefore \,\,\,\, x = 2$

Therefore, the value of $x$ is $2$ and it is the required solution for this problem.

Method: 2

The problem can also be solved in another method.

$3^{x+1} \times 5^{2x-1} = 3375$

Split the number $3375$ into multiplicative factors.

$3^{x+1} \times 5^{2x-1} = 3 \times 3 \times 3 \times 5 \times 5 \times 5$

Express same multiplicative factors in exponential form.

$3^{x+1} \times 5^{2x-1} = 3^3 \times 5^3$

Compare both sides of the equation. The base of first multiplicative terms of both sides of the equation is same and it is $3$. Similarly, the base of second multiplying terms of both sides is also equal. Therefore, $3^{x+1} = 3^3$ and $5^{2x-1} = 5^3$. So, their exponents are also equal mathematically.

Therefore $x+1 =3$ and $2x-1 = 3$.

$\implies x = 3-1$ and $2x = 3+1$.

$\implies x = 2$ and $2x = 4$.

$\implies x = 2$ and $x = \dfrac{4}{2}$.

Therefore $x = 2$ and $x = 2$.

The two equations have given same result. Therefore, the value of $x$ is $2$.

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