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. 4a^5(a^-1)^3/(a^-2)^-2

Accepted Answer

Hey everyone here we are asked to simplify the given expression where we have seven times K. To the power of eight times the quantity of K. To the power of negative two raised to the power of five. And this is all divided by the quantity of K. To the power of negative six raised to the power of negative six. So to simplify our given expression, we need to utilize the rules of exponents. And so the first rule of exponents that we need to recall is the power to a power rule, which tells us that the quantity of A. To the power of M. Raised to the power of N is equivalent to A. To the power of M times N. So now we can use this rule to rewrite our original expression. So beginning in our numerator we have seven times K. To the power of eight which remains the same. But the other part of our numerator can be rewritten using our rule. So we have K. Raised to the power of negative two times five. And so now we can move on to our denominator where we can also apply this rule. So we will have K raised to the power of negative six times negative six. And so now from here we just need to simplify our exponents. So in the numerator we again have seven times K. To the power of eight times K. To the power of negative 10. And this is divided by our denominator which simplifies two K. To the power of 36. So now our next step we need to recall another exponents rule which in this case is going to be the product rule which tells us that A to the power of M times A. To the power of N is equivalent to a to the power of em tie M plus N. So now we can use this rule to rewrite our expression. So we just need to rewrite the numerator in this case. So we have seven times K to the power of eight plus negative 10. And now our denominator remains the same for now. So we have K to the power of 36. And so now we just need to simplify the exponents in our numerator. So our expression becomes seven times K. To the power of eight plus negative 10, which just gives us negative two. And this is divided by K to the power of 36. So now we can move on to recalling our next rule of X opponents which now will be the quotient rule. While the quotient rule 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 here we can simplify our like base of K. Using this quotient rule. So our expression becomes seven times K. To the power of negative two because it because it's our power in the numerator minus the power in our denominator which is 36. And so now we just need to simplify this exponent here. So we have seven times K. To the power of negative 38. And now our last step here in simplifying is to recall the rule for a negative exponents which tells us that A. To the power of negative N is equivalent to one divided by A. To the power of positive N. So here we see our negative N. Is negative 38 which is the exponent for our K. Value here. So we can rewrite this expression as seven Divided by K. to the power of positive 38. And with this we have fully simplified our original expression. So we are left with answer choice. A 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. 4a^5(a^-1)^3/(a^-2)^-2

Key Concepts

Exponent Rules

Simplifying Expressions

Nonzero Variables

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