Calculate each value mentally. (0.1^3/2)(90^3/2)

Question

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression. We have parentheses 0.2 raised to the five thirds and parentheses times a new parentheses 135 raised to the five thirds and we are asked to compute the value cognitively. So we have to do this without a calculator. How do we do this? Well we're going to be recalling previous lessons about some of the rules for exponents. So I'm going to start off in this actually isn't related to a rule for exponents, but I'm going to rewrite this 0.2 as 1/5. So we've got 1/5 raised to the five thirds times 135 raised to the five thirds. And now the rule of exponents, we recall the quotient to power rule and that tells us that if a fraction is raised to an exponent then that exponent could be rewritten as being raised to both the numerator and the denominator. So that will be one raised to the five thirds over five, raised to the five thirds. And yes, this is still times 135 to the five thirds. But what's significant about this is that we can do the work on this one to the five thirds and simplify that. This is the same as taking the cubed root of one to the fifth, Power one to the fifth. Power is one and the cubed root of one is one. So this just equals one. So now our numerator going to give myself a little bit more room. Our numerator is 1/5 to the five thirds times 135 to the 5/3. And now we can just put that 135 to the five thirds over one. And we can multiply these two fractions, multiply enumerators straight across. So we now have 135 to the five thirds over five to the five thirds. And remember that quotient to power rule, we just said where that exponent gets raised to both the numerator and the denominator. Well here look we have the same exponent in our numerator and denominator. So we can take that and put it on the outside of the parentheses. We can rewrite this as 135 over five. All raised to the five thirds and now we can do that division inside the parentheses. 135 divided by five equals 27. So we have 27 raised to the five thirds. Well we could set that up as the cubed root of 27 to the fifth power but I don't know 27 to the fifth power off the top of my head. So we can actually do one more step to simplify this. We can think of 27 as three cubed three cubed is 27 that's still raised to the five thirds. We recall from previous lessons that when we have a power raised to a power we are going to multiply those exponents. So we have three raised to the three times five thirds, and those threes are going to cancel out their ones. So won over 51 times 5/1, 1 times five is five divided by one is still five. So this is three to the fifth. And that when we compute that three to the fifth is 243 we look at our answer choices and this matches with answer choice. D. Well done, we'll catch you on the next one.

Calculate each value mentally. (0.1^3/2)(90^3/2)

Key Concepts

Exponents

Order of Operations

Mental Math Techniques

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