Math  /  Algebra

Question23x4=1322^{3 x-4}=\frac{1}{32}

Studdy Solution

STEP 1

What is this asking? We need to find the value of xx that makes 22 raised to the power of 3x43x - 4 equal to 132\frac{1}{32}. Watch out! Remember, a fraction is just another way of writing a negative exponent!
Don't get tripped up by the fraction.

STEP 2

1. Rewrite the equation
2. Solve for xx

STEP 3

Alright, let's **rewrite** that fraction 132\frac{1}{32} as a power of **2**.
We know that 32=2532 = 2^5, right?
So, 132\frac{1}{32} is the same as 125\frac{1}{2^5}.

STEP 4

Now, remember the **negative exponent rule**: 1an=an\frac{1}{a^n} = a^{-n}.
Using this rule, we can rewrite 125\frac{1}{2^5} as 252^{-5}.
Boom!

STEP 5

So, our original equation 23x4=1322^{3x-4} = \frac{1}{32} becomes 23x4=252^{3x-4} = 2^{-5}.
Much better!

STEP 6

Since the **bases** are the same (both are **2**), we can set the **exponents** equal to each other.
This gives us the equation 3x4=53x - 4 = -5.

STEP 7

Now, let's **isolate** xx.
We want to get xx all by itself.
First, let's **add 4** to both sides of the equation: 3x4+4=5+43x - 4 + 4 = -5 + 4.
This simplifies to 3x=13x = -1.

STEP 8

Almost there!
Now, we just need to **divide** both sides by **3**: 3x3=13\frac{3x}{3} = \frac{-1}{3}.
This gives us our **final answer**: x=13x = -\frac{1}{3}.

STEP 9

x=13x = -\frac{1}{3}

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