Calculate each value mentally. (0.2^2/3)(40^2/3)

Question

Accepted Answer

Welcome back. I am so glad you're here. We are given this expression, parentheses. 0.125 raised to the 3/ and parentheses times new parentheses raised to the 3/5. And we are to compute this cognitively. So not using a calculator. How can we do that? Well we look at this and we recognize that 0.125 is the same thing as 1/8. So we could rewrite this as 1/8 raised to the 3/ times 256 raised to the 3/5. And now we can recall previous lessons on exponent rules and specifically the quotient to power rule, which tells us that this 3/5 can be distributed to both the numerator and the denominator. So we could rewrite this as one to the 3/5 over eight to the 3/5. And yes that is still times to the 3/5. So what we can do now that we have gone through the cushion to power rule is we can simplify what's in the numerator here. One to the 3/5 is the same as the fifth root of one to the third and one to the third equals one and the fifth root of one equals one. So our numerator is just one going to give myself a little bit more room. So we have one over eight to the 3/ times 256 to the 3/5. And now we can just put this 256 to the 3/5 over one and multiply straight across our numerator and denominators giving us 256 to the 3/5 in the numerator over eight to the 3/5 in the denominator. And now recall that quotient to power rule. Now we've got the same Exponent in the numerator and the denominator and we can pull those on the outside of parentheses. So we've got parentheses 256 divided by eight, all raised to the 3/5 and now we can do that 256 divided by eight in parentheses that equals 32. So we have 32 raised to the 3/5. Now again, that would translate to taking the fifth root of 32 to the third power, but I don't No 32 to the third power off the top of my head. So we can actually do some simplification here. We know that 32 is actually two to the fifth power and that is still raised to the 3/5. We recall from previous lessons that when we have a power raised to a power, those exponents are going to be multiplied by each other. So we're taking two raised to the 5th times 3/5. While those fives are going to cancel out, they'll become ones. So one times three which is three divided by one is still three. So we've got two raised to the third power here and two to the third equals eight. We look at our answer choices and that matches with answer choice. A well done, we'll catch you on the next one.

Calculate each value mentally. (0.2^2/3)(40^2/3)

Key Concepts

Exponents and Powers

Fractional Exponents

Order of Operations

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