In Exercises 93–102, solve each equation. 5^2x ⋅ 5^4x=125

Question

Accepted Answer

Welcome back. I am so glad you're here. We are given the equation four raised to the three X times four raised to the seven X equals 256. And we are asked to solve for X. Well, X is up here in the exponent. How do we get it out of the exponent? Let's see if we can do any simplification. We look here and we see that we have two terms with exponents and they have the same base. We recall from previous lessons on exponents that when we have two terms with the same base, they're exponents are going to be added together when the bases are multiplied. So we'll have four raised to the three X plus seven X equals 256 and three X plus seven X. Those are like terms we can combine them. So three X plus seven X is 10 X. So we have four raised to the 10 X equals. Now, this 256 can we simplify that? So that it has a base of four and is raised to an exponent as well? And yeah, 256 that's four to the fourth power. Now we have these two same bases, but they have different exponents, but they're equal. So what that means is that their exponents are equal to each other because those bases are the same. And we can write that we can write 10, X equals four and we can solve for X. Now we'll divide both sides by 10 and we get X equals 4/10 or because all of our answers are in decimals instead of fractions. That's 0.4. We look at our answer choices and that matches with answer choice. A well done. We'll catch you in the next one.

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In Exercises 93–102, solve each equation. 5^2x ⋅ 5^4x=125

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