Math  /  Algebra

QuestionTo solve 493x=3432x+149^{3 x}=343^{2 x+1}, write each side of the equation in terms of base \square DONE

Studdy Solution

STEP 1

What is this asking? We need to rewrite both sides of the equation 493x=3432x+149^{3x} = 343^{2x+1} using the same base. Watch out! Don't forget the exponent rules!
Especially when rewriting the bases and combining exponents.

STEP 2

1. Rewrite 49 and 343 with the same base.
2. Simplify exponents.
3. Solve for *x*.

STEP 3

Alright, let's **rewrite** these numbers!
Notice that both **49** and **343** are powers of **7**.
We have 49=7249 = 7^2 and 343=73343 = 7^3.
This is super helpful because it lets us rewrite the *entire* equation in terms of base **7**!

STEP 4

Substituting these into our original equation gives us (72)3x=(73)2x+1(7^2)^{3x} = (7^3)^{2x+1}.
See how we just swapped out the **49** for 727^2 and the **343** for 737^3?

STEP 5

Now, let's use our **exponent rules**!
Remember, (am)n=amn(a^m)^n = a^{m \cdot n}.
So, (72)3x(7^2)^{3x} becomes 723x=76x7^{2 \cdot 3x} = 7^{6x}.

STEP 6

Similarly, (73)2x+1(7^3)^{2x+1} becomes 73(2x+1)=76x+37^{3(2x+1)} = 7^{6x+3}.
Don't forget to distribute the **3** to *both* terms in the parentheses!

STEP 7

Our equation now looks like this: 76x=76x+37^{6x} = 7^{6x+3}.
Much cleaner, right?

STEP 8

Since the bases are the same, the exponents *must* be equal!
This gives us the equation 6x=6x+36x = 6x + 3.

STEP 9

Let's **isolate** *x*.
Subtract 6x6x from both sides of the equation.
This gives us 6x6x=6x+36x6x - 6x = 6x + 3 - 6x, which simplifies to 0=30 = 3.

STEP 10

Uh oh! 0=30 = 3 is *never* true.
This means there's **no solution** for *x* that will make the original equation true.

STEP 11

There is no solution for *x*.

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