Question Solve for $y$ in the equation $5 y^{\frac{2}{3}} + 7 = 27$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $5 y^{\frac{2}{3}} + 7 = 27$ .
2. We need to solve for the variable $y$ .

STEP 2

First, we need to isolate the term containing the variable $y$ on one side of the equation. We can do this by subtracting $7$ from both sides of the equation.
$5 y^{\frac{2}{3}} + 7 - 7 = 27 - 7$

STEP 3

Simplify both sides of the equation.
$5 y^{\frac{2}{3}} = 20$

STEP 4

Next, we need to isolate $y^{\frac{2}{3}}$ . We can do this by dividing both sides of the equation by $5$ .
$\frac{5 y^{\frac{2}{3}}}{5} = \frac{20}{5}$

STEP 5

Simplify both sides of the equation.
$y^{\frac{2}{3}} = 4$

STEP 6

To solve for $y$ , we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal power of $\frac{2}{3}$ , which is $\frac{3}{2}$ .
$\left(y^{\frac{2}{3}}\right)^{\frac{3}{2}} = 4^{\frac{3}{2}}$

STEP 7

When we raise a power to a power, we multiply the exponents. The left side simplifies because $\frac{2}{3} \times \frac{3}{2} = 1$ , which leaves us with $y$ to the power of $1$ , or simply $y$ .
$y = 4^{\frac{3}{2}}$

STEP 8

To calculate $4^{\frac{3}{2}}$ , we can first take the square root of $4$ and then raise the result to the power of $3$ .
$4^{\frac{1}{2}} = \sqrt{4} = 2$

STEP 9

Now, raise $2$ to the power of $3$ .
$2^3 = 2 \times 2 \times 2 = 8$

STEP 10

Thus, we find the value of $y$ .
$y = 8$
The solution to the equation $5 y^{\frac{2}{3}} + 7 = 27$ is $y = 8$ .

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Question Solve for $y$ in the equation $5 y^{\frac{2}{3}} + 7 = 27$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $5 y^{\frac{2}{3}} + 7 = 27$ .
2. We need to solve for the variable $y$ .

STEP 2

First, we need to isolate the term containing the variable $y$ on one side of the equation. We can do this by subtracting $7$ from both sides of the equation.
$5 y^{\frac{2}{3}} + 7 - 7 = 27 - 7$

STEP 3

Simplify both sides of the equation.
$5 y^{\frac{2}{3}} = 20$

STEP 4

Next, we need to isolate $y^{\frac{2}{3}}$ . We can do this by dividing both sides of the equation by $5$ .
$\frac{5 y^{\frac{2}{3}}}{5} = \frac{20}{5}$

STEP 5

Simplify both sides of the equation.
$y^{\frac{2}{3}} = 4$

STEP 6

To solve for $y$ , we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal power of $\frac{2}{3}$ , which is $\frac{3}{2}$ .
$\left(y^{\frac{2}{3}}\right)^{\frac{3}{2}} = 4^{\frac{3}{2}}$

STEP 7

When we raise a power to a power, we multiply the exponents. The left side simplifies because $\frac{2}{3} \times \frac{3}{2} = 1$ , which leaves us with $y$ to the power of $1$ , or simply $y$ .
$y = 4^{\frac{3}{2}}$

STEP 8

To calculate $4^{\frac{3}{2}}$ , we can first take the square root of $4$ and then raise the result to the power of $3$ .
$4^{\frac{1}{2}} = \sqrt{4} = 2$

STEP 9

Now, raise $2$ to the power of $3$ .
$2^3 = 2 \times 2 \times 2 = 8$

STEP 10

Thus, we find the value of $y$ .
$y = 8$
The solution to the equation $5 y^{\frac{2}{3}} + 7 = 27$ is $y = 8$ .

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Question Solve for yyy in the equation 5y23+7=275 y^{\frac{2}{3}} + 7 = 275y32​+7=27.

Studdy Solution

STEP 1

Assumptions1. We are given the equation 5y23+7=275 y^{\frac{2}{3}} + 7 = 275y32​+7=27.2. We need to solve for the variable yyy.

STEP 2

First, we need to isolate the term containing the variable yyy on one side of the equation. We can do this by subtracting 777 from both sides of the equation.5y23+7−7=27−75 y^{\frac{2}{3}} + 7 - 7 = 27 - 75y32​+7−7=27−7

STEP 3

Simplify both sides of the equation.5y23=205 y^{\frac{2}{3}} = 205y32​=20

STEP 4

Next, we need to isolate y23y^{\frac{2}{3}}y32​. We can do this by dividing both sides of the equation by 555.5y235=205\frac{5 y^{\frac{2}{3}}}{5} = \frac{20}{5}55y32​​=520​

STEP 5

Simplify both sides of the equation.y23=4y^{\frac{2}{3}} = 4y32​=4

STEP 6

To solve for yyy, we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal power of 23\frac{2}{3}32​, which is 32\frac{3}{2}23​.(y23)32=432\left(y^{\frac{2}{3}}\right)^{\frac{3}{2}} = 4^{\frac{3}{2}}(y32​)23​=423​

STEP 7

When we raise a power to a power, we multiply the exponents. The left side simplifies because 23×32=1\frac{2}{3} \times \frac{3}{2} = 132​×23​=1, which leaves us with yyy to the power of 111, or simply yyy.y=432y = 4^{\frac{3}{2}}y=423​

STEP 8

To calculate 4324^{\frac{3}{2}}423​, we can first take the square root of 444 and then raise the result to the power of 333.412=4=24^{\frac{1}{2}} = \sqrt{4} = 2421​=4​=2

STEP 9

Now, raise 222 to the power of 333.23=2×2×2=82^3 = 2 \times 2 \times 2 = 823=2×2×2=8

STEP 10

Thus, we find the value of yyy.y=8y = 8y=8The solution to the equation 5y23+7=275 y^{\frac{2}{3}} + 7 = 275y32​+7=27 is y=8y = 8y=8.

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Question Solve for $y$ in the equation $5 y^{\frac{2}{3}} + 7 = 27$ .

Assumptions
1. We are given the equation $5 y^{\frac{2}{3}} + 7 = 27$ .
2. We need to solve for the variable $y$ .

First, we need to isolate the term containing the variable $y$ on one side of the equation. We can do this by subtracting $7$ from both sides of the equation.
$5 y^{\frac{2}{3}} + 7 - 7 = 27 - 7$

Simplify both sides of the equation.
$5 y^{\frac{2}{3}} = 20$

Next, we need to isolate $y^{\frac{2}{3}}$ . We can do this by dividing both sides of the equation by $5$ .
$\frac{5 y^{\frac{2}{3}}}{5} = \frac{20}{5}$

Simplify both sides of the equation.
$y^{\frac{2}{3}} = 4$

To solve for $y$ , we need to get rid of the fractional exponent. We can do this by raising both sides of the equation to the reciprocal power of $\frac{2}{3}$ , which is $\frac{3}{2}$ .
$\left(y^{\frac{2}{3}}\right)^{\frac{3}{2}} = 4^{\frac{3}{2}}$

When we raise a power to a power, we multiply the exponents. The left side simplifies because $\frac{2}{3} \times \frac{3}{2} = 1$ , which leaves us with $y$ to the power of $1$ , or simply $y$ .
$y = 4^{\frac{3}{2}}$

To calculate $4^{\frac{3}{2}}$ , we can first take the square root of $4$ and then raise the result to the power of $3$ .
$4^{\frac{1}{2}} = \sqrt{4} = 2$

Now, raise $2$ to the power of $3$ .
$2^3 = 2 \times 2 \times 2 = 8$

Thus, we find the value of $y$ .
$y = 8$
The solution to the equation $5 y^{\frac{2}{3}} + 7 = 27$ is $y = 8$ .