QuestionUse the equation $2,401=7^{6-2 x}$ . What power of 7 is 2,401 ? 3 4 5 DONE

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes $2401$ equal to $7$ raised to the power of $6 - 2x$ . Watch out! Don't get tricked by the $6 - 2x$ !
We need to treat it as one single unit until we isolate $x$ .

STEP 2

1. Rewrite 2401 as a power of 7
2. Set the exponents equal
3. Isolate $x$

STEP 3

Let's break down 2401 into its prime factors.
We're doing this to see how many times 7 multiplies by itself to make 2401.
This will help us rewrite 2401 as $7$ to some power.

STEP 4

$2401 \div 7 = 343$ Great! 7 is a factor.
Let's keep going!

STEP 5

$343 \div 7 = 49$ Awesome! Another 7!

STEP 6

$49 \div 7 = 7$ And finally, $7 \div 7 = 1$ Perfect! We've reached 1, so we're done dividing.

STEP 7

We divided by 7 a total of four times.
This means $2401 = 7 \cdot 7 \cdot 7 \cdot 7 = 7^4$ .
So, 2401 is $7$ to the fourth power!

STEP 8

We know $2401 = 7^4$ , and our original equation is $2401 = 7^{6 - 2x}$ .
So, we can rewrite the original equation as: $7^4 = 7^{6 - 2x}$

STEP 9

Since the bases are the same (both are 7), the exponents must be equal for the equation to be true.
This is a crucial step!
So, we get: $4 = 6 - 2x$

STEP 10

We want to get $x$ by itself.
Let's start by subtracting 6 from both sides of the equation: $4 - 6 = 6 - 2x - 6$ $-2 = -2x$

STEP 11

Now, let's divide both sides by $-2$ to isolate $x$ : $\frac{-2}{-2} = \frac{-2x}{-2}$ $1 = x$ So, $x = \mathbf{1}$ !

STEP 12

$x = 1$ makes the original equation true. $7^{6 - 2(1)} = 7^{6 - 2} = 7^4 = 2401$ .
The power of 7 that equals 2401 is 4.

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