What values of c and d make the equation true ∛(162x^c y⁵) = 3x²y(∛6y^d)

Solution:

Let us use the exponents rule to find the value of c and d.

Given: ∛(162x^c y⁵) = 3x²y (∛6y^d)

By taking cube on both sides

( ∛(162x^c y⁵))³ = ( 3x²y(∛6y^d))³

⇒ 162x^c y⁵ = ( 3x²y)³( ∛6y^d)³

By using the exponents rule (a^m)ⁿ = a^mn

⇒ 162x^c y⁵ = 27 x⁶y³ × 6y^d

By using the exponents rule a^m × aⁿ = a^{m + n}

⇒ 162x^c y⁵ = 162 x⁶y^{3 + d}

By comparing the powers of the corresponding variables on both the sides we get

c= 6; 5 = 3 + d or d = 2

What values of c and d make the equation true ∛(162x^c y⁵) = 3x²y(∛6y^d)

Summary:

The values of c and d that make the equation ∛(162x^c y⁵) = 3x²y(∛6y^d) true are 6 and 2 respectively.