To solve the equation \(16^{y^3} = \sqrt[3]{4}^x\) by making the bases common, we can express both sides of the equation with a common base.
First, let's rewrite the bases:
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\(16\) can be rewritten as \(2^4\): \[ 16^{y^3} = (2^4)^{y^3} = 2^{4y^3} \]
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For \(\sqrt[3]{4}\), we can express \(4\) as \(2^2\): \[ \sqrt[3]{4} = \sqrt[3]{2^2} = 2^{\frac{2}{3}} \] Now, raising this to the power of \(x\): \[ \left(\sqrt[3]{4}\right)^x = \left(2^{\frac{2}{3}}\right)^x = 2^{\frac{2x}{3}} \]
Now we can rewrite the equation: \[ 2^{4y^3} = 2^{\frac{2x}{3}} \]
Since the bases are the same, we can set the exponents equal to each other: \[ 4y^3 = \frac{2x}{3} \]
To eliminate the fraction, we can multiply both sides by 3: \[ 3(4y^3) = 2x \] \[ 12y^3 = 2x \]
Now, divide both sides by 2: \[ 6y^3 = x \]
Thus, the relationship between \(x\) and \(y\) is: \[ x = 6y^3 \]