Questions: Реши неравенство 8^(x/3) < 64^(x+7/2).

Solution Steps

To solve the inequality \(8^{\frac{x}{3}} < 64^{\frac{x+7}{2}}\), we can start by expressing both sides with the same base. Notice that \(8 = 2^3\) and \(64 = 2^6\). This allows us to rewrite the inequality in terms of base 2. Then, we can simplify the exponents and solve for \(x\).

Given the inequality \(8^{\frac{x}{3}} < 64^{\frac{x+7}{2}}\), we start by expressing both sides with the same base. Notice that \(8 = 2^3\) and \(64 = 2^6\). Thus, we can rewrite the inequality as: \[ (2^3)^{\frac{x}{3}} < (2^6)^{\frac{x+7}{2}} \]

Next, we simplify the exponents: \[ 2^x < 2^{3(x+7)} \] This simplifies to: \[ 2^x < 2^{3x + 21} \]

Since the bases are the same, we can compare the exponents: \[ x < 3x + 21 \] Subtract \(x\) from both sides: \[ 0 < 2x + 21 \] Subtract 21 from both sides: \[ -21 < 2x \] Divide by 2: \[ -\frac{21}{2} < x \]

Final Answer