Question

15^2x-3 = 3^x * 5^3x-6

Answer 1

Х=4/15, х=0,266667 !!!

Answer 2

To solve the equation $15^{2x-3} = 3^x \times 5^{3x-6}$, we can start by simplifying both sides of the equation using the properties of exponents. Let's break down the steps:

Step 1: Express both sides in terms of prime factors. $15 = 3 \times 5$ and $3^2 = 9$, so $15^2 = (3 \times 5)^2 = 3^2 \times 5^2 = 9 \times 25$. Similarly, $5^3 = 5 \times 5 \times 5$.

Step 2: Rewrite the equation with the new expressions. $(9 \times 25)^{x-3} = 3^x \times (5 \times 5 \times 5)^{x-6}$.

Step 3: Apply the properties of exponents. Using the rule $(a \times b)^n = a^n \times b^n$, we can rewrite the equation as follows: $9^{x-3} \times 25^{x-3} = 3^x \times 5^{3(x-6)}$.

Step 4: Further simplify the equation. $9^{x-3}$ can be rewritten as $(3^2)^{x-3} = 3^{2(x-3)} = 3^{2x-6}$. $25^{x-3}$ can be rewritten as $(5^2)^{x-3} = 5^{2(x-3)} = 5^{2x-6}$.

The equation becomes: $3^{2x-6} \times 5^{2x-6} = 3^x \times 5^{3(x-6)}$.

Step 5: Use the rule $a^{m} \times a^{n} = a^{m+n}$ to combine exponents with the same base. $3^{2x-6 + x} \times 5^{2x-6 + 3(x-6)} = 3^x \times 5^0$.

Step 6: Simplify further. $3^{3x-6} \times 5^{2x-6 + 3x-18} = 3^x$.

Step 7: More simplification. $3^{3x-6} \times 5^{5x-24} = 3^x$.

Step 8: Now, to solve for $x$, we can equate the exponents of $3$ on both sides since the bases are the same.

$3x - 6 = x$.

Step 9: Solve for $x$.

Subtract $x$ from both sides: $3x - x = 6$. $2x = 6$.

Divide by $2$: $x = \frac{6}{2}$.

$x = 3$.

So, the value of $x$ that satisfies the equation is $x = 3$.