To express the equation \(16y^3 = \left(\sqrt[3]{4}\right)^x\) with a common base, we first need to express both sides in terms of powers of the same base.
Starting with the left side:
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Convert 16 into a power of 4: \[ 16 = 4^2 \] So, we rewrite the left side: \[ 16y^3 = 4^2 y^3 = 4^2 \cdot y^3 \]
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Now, focusing on the right side: The right side is already expressed as \(\left(\sqrt[3]{4}\right)^x\). We can convert this into a base of 4: \[ \sqrt[3]{4} = 4^{1/3} \] Therefore, \[ \left(\sqrt[3]{4}\right)^x = \left(4^{1/3}\right)^x = 4^{x/3} \]
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Now we can set the equation with the common base: Substitute back into the original equation: \[ 4^2 \cdot y^3 = 4^{x/3} \]
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Equate the exponents: Since both sides are powers of 4, we can equate the exponents: \[ 2 + \log_4(y^3) = \frac{x}{3} \] Or simplifying \(\log_4(y^3)\): \[ \log_4(y^3) = 3 \log_4(y) \]
So the equation can be rewritten: \[ 2 + 3 \log_4(y) = \frac{x}{3} \]
Therefore, you could express the final equation with a common base and equate the exponents.