In Exercises 55–78, use properties of rational exponents to simpl...

Question

In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.3^¾ ⋅ 3^¼

Accepted Answer

Hey, everyone in this problem, we're asked to simplify the following expression by using the properties of rational exponents. The expression we're given is 102 to the exponent 4/7 multiplied by 102 to the exponent 3 7th. We're given four answer choices. Option A one, option B 102. Option C 102 to the exponent divided by 49 and ad 102 to the exponent 1/7. We're gonna start by rewriting the expression we were given. So we have 102 to the exponent 4 7th multiplied by 102 to the exponent 3 7th. Now we have two things multiplied together that have the same base. So let's recall our property of rational exponent that allows us to simplify something like this. If we have X to the exponent M multiplied by X to the exponent N, we can write this as X to the exponent M plus X. OK. So two things multiplied together, we can simplify them into a single term. We keep the same base and we add the original exponents together to get our new exponent. If we apply this to our case, we get 102 to the exponent 4/ plus three sevens. So now we've simplified this into a single term. Yeah, we wanna go ahead and simplify that exponent. We're adding two fractions. They already have the same denominator. So we can go ahead and add, we have four sevens plus three sevens. This is gonna give us an exponent of seven divided by seven. So we have 102 to the exponent seven divided by seven and well, seven divided by seven is just equal to one. So we get 102 to the exponent one. And we know that we don't have to write that exponent one that's implied. And so we can write this as just 102. So the original expression we were given simplifies down to just the value 102. So the correct answer is option B- 102. Thanks everyone for watching. I hope this video helped see you in the next one.

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In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.3^¾ ⋅ 3^¼

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