Simplify each expression. Write answers without negative exponent...

Question

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (5x)^-2(5x^3)^-3/(5^-2x^-3)^-3

Accepted Answer

Hey everyone here we are asked to simplify the following expression where we have the quantity of two times Y raised to the power of negative three times the quantity of two times Y raised to the fourth power raised to the power of negative two. And this is all divided by the quantity of two raised to the power of negative three times Y raised to the the power of negative to all raised to the power of two. So to simplify this given expression here, we need to utilize our rules of exponents. And so the first rules of exponents that we need to recall is the power rule of a product. Which tells us that the quantity of A times B raised to the power of M is equivalent to a raised to the power of M times B raised to the power of em. So now utilizing this rule here we can rewrite our original expression. So beginning with our first term in our numerator, we see that the outside exponents. Is this negative three. So we can go ahead and just distribute it into this first term. So doing this, we have the quantity of two raised to the power of negative three times Y raised to the power of negative three. And now moving on to our next term in our numerator here we need to distribute in this negative two. So doing this, we have the quantity of two raised to the power of negative two times the quantity of why raised to the power power of four raised to the power of negative two. And now moving on to our denominator here, we see, we need to distribute in this positive two exponents. So doing this, we have the quantity of two raised to the power of negative three, raised to the power of two times the quantity of why raised to the power of negative two raised to the power of two. And so now we need to recall our next exponent rule which now is the rule for a power to a power. And so this rule tells us that the quantity of a. To the power of N raised to the power of M is equivalent to a. To the power of N. Times M. So now with this rule in mind we can go ahead and again rewrite our expression here. So our first term in the numerator is just going to remain the same as two raised to the power of negative three times Y raised to the power of negative three. And now this first term is multiplied by the quantity of two raised to the power of negative two. And now our next part of our second term here, our next y term. We see that we can apply our power to a power rule. So our Y term here, we have y raised to the power of four times four times negative two. And so now we can go ahead and move on to our denominator here where we see we can apply this rule here to both of the terms. So we have two raised to the power of negative three times two times Y raised to the power of negative two times two. And so now we can just simplify our exponents here. So our expression becomes two raised to the power of negative three times Y raised to the power of negative three times the quantity of two raised to the power of negative two times Y raised to the power of negative eight. And this expression here is divided by our deny dominator which simplifies to to raise to the power of negative six times why? Raised to the power of negative four. And so now we are ready to recall our next exponent rule which will now be the product rule for exponents which tells us that A. To the power of mm times A. To the power of N is equivalent to a. To the power of M plus N. So now we can apply this rule to our expression that we have so far and we see we can begin with our numerator here. So we need to combine our like bases. So we can start with the like base of two. So we have two raised to the power of negative three plus the other power of two which is negative two. And now this first term is multiplied by the Y term which now simplifies to Y. To the power of negative three plus negative eight. And now moving on to our denominator in this case it just stays the same for now. So we have two raised to the power of negative six times Y raised to the power of of negative four. So now we can simplify our exponents that we have in the numerator. So we have two raised to the power of negative five times Y raised to the power of negative 11. And again this is divided by the quantity of two raised to the power of negative six times Y raised to the power of negative four. So now we can move on to recalling our next um exponents rule which now is going to be the quotient rule which tells us that A to the power of m divided by a to the power of N is equivalent to a. To the power of M minus n. So once again we need to combine our like basis here from our expression with our formula. So doing this, our expression becomes starting with our to base, we have two raised to the power of negative five because it is the exponent in the numerator minus the exponents in the denominator which is negative six. So we have two raised to the power of negative five minus negative six. And now moving on to our Y term, we have y raised to the power of negative minus negative four. So now once again we just have to simplify our exponents that we have so far. And doing this we have two raised to the power of negative five minus negative six which just gives us two raised to the power of one. And then this is times Y. Which again is raised to the power of negative 11 minus negative four which gives us negative seven. So to now again we can simplify our expression. So we have just two times why raised to the power of negative seven. And so now we can recall our last exponent rule which is going to be the rule for a negative exponents. And this tells us that a. To the power of a. To the power of negative N. Is equivalent to one divided by a. To the power of N. So once again we need to apply this rule to the expression which we have found so far. So again we have the expression two times y raised to the power of negative seven. And now using our rule we can rewrite this expression as two divided by why raised to the power of positive seven. And with this we have fully simplified our original expression and we are left with answer choice D Thanks for watching. I hope you found this video helpful and I'll see you again next time

Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (5x)^-2(5x^3)^-3/(5^-2x^-3)^-3

Key Concepts

Negative Exponents

Exponent Rules

Simplifying Rational Expressions

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