Questions: Solve the following equation. If needed, submit your answer as a fraction reduced to lowest terms. y^(2/3) - 25/64 = 0

$Solve the following equation. If needed, submit your answer as a fraction reduced to lowest terms. y^(2/3) - 25/64 = 0$

Transcript text: Solve the following equation. If needed, submit your answer as a fraction reduced to lowest terms. \[ y^{\frac{2}{3}}-\frac{25}{64}=0 \]

Solution

Solution Steps

Step 1: Isolate the Variable

Starting with the equation: \[ y^{\frac{2}{3}} - \frac{25}{64} = 0 \] we isolate the variable term by adding $\frac{25}{64}$ to both sides: \[ y^{\frac{2}{3}} = \frac{25}{64} \]

Step 2: Raise to the Reciprocal Power

Next, we raise both sides of the equation to the reciprocal of the fractional exponent, which is $\frac{3}{2}$: \[ \left(y^{\frac{2}{3}}\right)^{\frac{3}{2}} = \left(\frac{25}{64}\right)^{\frac{3}{2}} \] This simplifies to: \[ y = \left(\frac{25}{64}\right)^{\frac{3}{2}} \]

Step 3: Calculate the Right Side

Calculating $\left(\frac{25}{64}\right)^{\frac{3}{2}}$: \[ \left(\frac{25}{64}\right)^{\frac{3}{2}} = \frac{25^{\frac{3}{2}}}{64^{\frac{3}{2}}} = \frac{125}{512} \]