Question2. $2(x+3)^{\frac{3}{2}}=54$

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes this equation true! Watch out! Remember the order of operations—we'll need to carefully undo each operation one by one to isolate $x$ .
Don't forget about those fractional exponents!

STEP 2

1. Isolate the term with $x$ .
2. Undo the exponent.
3. Solve for $x$ .

STEP 3

We have $2(x+3)^{\frac{3}{2}}=54$ .
To isolate the term with $x$ , $(x+3)^{\frac{3}{2}}$ , we need to divide both sides of the equation by the coefficient $2$ .
This gives us $(x+3)^{\frac{3}{2}} = \frac{54}{2} = 27$ .
Awesome!

STEP 4

Now we have $(x+3)^{\frac{3}{2}} = 27$ .
To get rid of that $\frac{3}{2}$ exponent, we can raise both sides to the reciprocal power, which is $\frac{2}{3}$ .
Remember, when we raise a power to another power, we multiply the exponents.
So, $\left((x+3)^{\frac{3}{2}}\right)^{\frac{2}{3}} = (x+3)^{\frac{3}{2} \cdot \frac{2}{3}} = (x+3)^1 = x+3$ .
On the other side, we have $27^{\frac{2}{3}}$ .

STEP 5

Remember that $27^{\frac{2}{3}}$ is the same as $\left(27^{\frac{1}{3}}\right)^2$ .
The cube root of $27$ is $3$ , so we have $3^2$ , which is $9$ .
So, our equation becomes $x+3 = 9$ .
We're almost there!

STEP 6

We're so close!
We have $x+3 = 9$ .
To isolate $x$ , we need to subtract $3$ from both sides.
This gives us $x + 3 - 3 = 9 - 3$ , which simplifies to $x = 6$ .
Boom!

STEP 7

$x = 6$

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