Clarification of ‘term’ in raising power to a power?

Question

According to the text-book, ‘when there is more than one term inside a bracket all of the terms must be taken to the power outside the bracket’

In the following example:

[6a^(5t)/3a^(3t)]^4

The expression in brackets simplifies to:

[2a^(2t)]^4

=2^(4)*a^(2t)^4
=16a^(8t)

So in this case the coefficient of a is considered a term and 2t is considered a term but a isn’t?
I can do the examples in the exercise,it’s the definition of term that is giving me the problem. I’d be grateful if someone could clear this up for me,

Answer 1

yes, it's right



your textbook is not much precise

it must be said in this way

‘when there is more than one factor inside a bracket all of the factor must be taken to the power outside the bracket’



that is



[ab]^n = a^nb^n



and also

[a/b]^n = a^n/b^n



but applying it to sums or differences is an error



(2 + 3)^2 is different from 2^2 + 3^2

the first is 25, the second is 13



So your question is very important



to calculate

[2a^(2t)]^4

you 'distribute the power 4 on the factors



(2^4)(a^(2t))^4



2t is definitely NOT a factor.

You can go on applying powers properties as you did



16 a^(8t)



and that's all

Answer 2

The word term is the source of confusion.**

In a fraction, the numerator is one term and the denominator is the other term.



So all the book is saying is:



[6a^(5t)/3a^(3t)]^4



becomes



[6 a^(5t)]^4 / [3a^(3t)]^4



The explanation about "terms" stops here. It does not tell you how it works inside each bracket.

The "things" inside the brackets are factors (objects that are multiplied one with the other).



When there is a power outside of a bracket containing many factors, then all factors are raised to that power



(ab/c)^3 = (ab)^3 / c^3 = (a^3)(b^3)/(c^3)



In your example:



[6a^(5t)/3a^(3t)]^4



The factors are:

above the line

6 and a^(5t)

("a" to the power 5t)

a^(5t) is one factor (5t is not a factor, it is a power)



below the line

3 and a^(3t)



You can simplify before applying the outside power (as you did, which is the easier way to do it) or you can do it formally: apply the outside power to all factors, then simplify:



(6^4) [a^(5t)]^4 / {(3^4) [a^(3t)]^4}



One at a time:

6^4 = 1296

[a^(5t)]^4 = a^(20t)

when raising a power to a power, multiply the powers

3^4 = 81

[a^(3t)]^4 = a^(12t)

This comes from the multiplication rule: when multiplying powers of a base, add the powers:

[a^(3t)]^4 = a^(3t) * a^(3t) * a^(3t) * a^(3t) =a^(3t+3t+3t+3t) = a^(12t)

In this manner, we did apply the ^4 to a, through its already existing power.



Rewrite the whole thing

[1296 a^(20t)] / [81 a^(12t)

= (1296/81) [a^(20t) / a^(12t)]

= 16 a^(20t - 12t) = 16 a^(8t)



---



In mathematics, a "term" is an element of a sum or a difference.



example:

6 + a^t

Here, 6 is a term and a^t is another term



But, in the expression

6a^t (where they are multiplied together)

there is only one term: 6a^t





and when a bracket containing many "terms" is raised to a power, the contents of the bracket are multiplied by everything in the bracket:



(6 + a^t)^2 = (6 + a^t)(6 + a^t)

=6(6 + a^t) + a^t(6 + a^t)

= 36 + 6a^t + 6a^t + a^(2t)

=36 + 12a^t + a^(2t)



which is NOT the same as 6^2 + (a^t)^2

Answer 3

No. The terms in



[6a^(5t)/3a^(3t)]^4



are 6a^(5t) and 3a^(3t) ----- only 2 terms. And both terms are of the variable a.



6 is a coefficient, and 5t is an exponent. But they are both parts of the term 6a^(5t)



[ In expressions with several variables, you might see reference to "the term in x, or the term in y, or whatever. In the case of the terms here, "the term in a" could refer to 6a^5t, or to 3a^3t, but it means the whole thing; 6 would never be called a 'term', nor would 5t. ]





Similarly, 3a^(3t) is also a term in the variable a.



The first 3 is a coefficient, the second 3 is (part of) the exponent



So, raising 'everything' to the power 4 means just the two terms, thus



(6a^(5t)) ^4 . . . . . . 6^4 ∙ (a^5t)^4

(---------) . . . = . . . ------------------

(3a^(3t)) . . . . . . . . .3^4 ∙ (a^3t)^4



(The coefficients 6 and 3 are only a part of each term, but it is necessary to raise the whole term to the power, therefore they must be raised to the power along with the rest of the term.)





And raising a power to a power, as in (a^5t)^4 means multiply the powers, so = a^20t.



Similarly (a^3t)^4 = a^12t (but the term being raised is a^3t, not merely a, and certainly not 3t)





Therefore



(6a^(5t)) ^4 . . . . . . 1296 ∙ a^20t

(---------) . . . = . . . . .----------------- . = . 16 a^8t, as you correctly observe.

(3a^(3t)) . . . . . . . . .. 81 ∙ a^12t

Answer 4

yes you are totally right. your textbook has something missing thus confusing you

Answer 5

what you wrote is OK . It is 16a^(8t)