BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.3, Problem 135E

  (a)

To determine

To calculate: The factors of the expression A4B4 and A6B6 .

Expert Solution

Answer to Problem 135E

The factors of the expression A4B4=(AB)(A+B)(A2+B2)

  and A6B6=(AB)(A2+B2+AB)(A+B)(A2+B2AB) .

Explanation of Solution

Given information:

The expression A4B4 and A6B6 .

Formula used:

The difference of square of two numbers a and b is a2b2=(a+b)(ab) .

Calculation:

Consider the expression A4B4 and A6B6 .

Recall that the difference of square of two numbers a and b is a2b2=(a+b)(ab) .

Apply it,

  A4B4=(A2)2(B2)2=(A2B2)(A2+B2)

Again apply the difference of square of two numbers,

  A4B4=(A2B2)(A2+B2)=(AB)(A+B)(A2+B2)

And,

  A6B6=(A3)2(B3)2=(A3B3)(A3+B3)

Again apply the difference and sum of cube of two numbers,

  A6B6=(A3)2(B3)2=(A3B3)(A3+B3)=(AB)(A2+B2+AB)(A+B)(A2+B2AB)

Thus, the factors of the expression A4B4=(AB)(A+B)(A2+B2)

  and A6B6=(AB)(A2+B2+AB)(A+B)(A2+B2AB) .

  (b)

To determine

To verify: The values of the expression 18335=12474 and 2868335=12676 .

Expert Solution

Explanation of Solution

Given information:

The expression 18335=12474 and 2868335=12676 .

Formula used:

The difference of fourth power of two numbers A and B is A4B4=(AB)(A+B)(A2+B2) and difference of sixth power A6B6=(AB)(A2+B2+AB)(A+B)(A2+B2AB) .

Calculation:

Consider the expression 18335=12474 and 2868335=12676 .

Recall that the difference of fourth power of two numbers A and B is A4B4=(AB)(A+B)(A2+B2) and difference of sixth power A6B6=(AB)(A2+B2+AB)(A+B)(A2+B2AB) .

Apply it,

  12474=(127)(12+7)(122+72)=5(19)(144+49)=5(19)(193)=18335

And,

  12676=(127)(12+7)(122+72+12(7))(122+7212(7))=5(19)(144+49+84)(144+4984)=5(19)(277)(109)=2868335

Hence, it is verified that values of the expression 18335=12474 and 2868335=12676 .

  (c)

To determine

To show: The factors 18335 and 2868335 are prime numbers.

Expert Solution

Explanation of Solution

Given information:

The expression 18335=12474 and 2868335=12676 .

Formula used:

The difference of fourth power of two numbers A and B is A4B4=(AB)(A+B)(A2+B2) and difference of sixth power A6B6=(AB)(A2+B2+AB)(A+B)(A2+B2AB) .

Calculation:

Consider the expression 18335=12474 and 2868335=12676 .

Recall that the difference of fourth power of two numbers A and B is A4B4=(AB)(A+B)(A2+B2) and difference of sixth power A6B6=(AB)(A2+B2+AB)(A+B)(A2+B2AB) .

Apply it,

  18335=12474=(127)(12+7)(122+72)=5(19)(144+49)=5(19)(193)

Therefore, factors of 18335 are 5, 19 and 193 which are all prime numbers.

And,

  2868335=12676=(127)(12+7)(122+72+12(7))(122+7212(7))=5(19)(144+49+84)(144+4984)=5(19)(277)(109)

Therefore, factors of 2868335 are 5, 19, 277 and 109 which are all prime numbers.

Hence, it is shown that factors 18335 and 2868335 are prime numbers.

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