Question

write the following expression using exponents:
5×5×d×5×d×5×d
part a
the exponent for 5 is
part b
the exponent for d is

Explanation:

Step1: Count occurrences of 5

In the expression \(5\times5\times d\times5\times d\times5\times d\), count how many times 5 appears. Let's list the factors: 5, 5, d, 5, d, 5, d. The number of 5s is 4 (first 5, second 5, fourth 5, sixth 5? Wait, no, let's count again: positions 1:5, 2:5, 4:5, 6:5? Wait no, let's list all terms: 5 (1), 5 (2), d (1), 5 (3), d (2), 5 (4), d (3). So 5 appears 4 times? Wait no, 1st term 5, 2nd 5, 4th 5, 6th 5? Wait no, the expression is \(5\times5\times d\times5\times d\times5\times d\). Let's count the 5s: first 5, second 5, fourth term 5, sixth term 5? Wait no, let's write the terms in order:

Term 1: 5

Term 2: 5

Term 3: d

Term 4: 5

Term 5: d

Term 6: 5

Term 7: d

So number of 5s: term1, term2, term4, term6 → that's 4? Wait no, 1,2,4,6: four times? Wait 1,2,4,6: 4? Wait 1 (5), 2 (5), 4 (5), 6 (5): yes, 4 times. Wait no, wait 1,2,4,6: that's four 5s? Wait 1,2,4,6: 4? Wait 1 (5), 2 (5), 4 (5), 6 (5): yes, 4. Wait but let's check again: the expression is \(5\times5\times d\times5\times d\times5\times d\). Let's group the 5s: 5×5×5×5 (since there are four 5s? Wait no, 5 appears in positions 1,2,4,6? Wait no, 1,2,4,6: that's four 5s? Wait 1 (5), 2 (5), 4 (5), 6 (5): yes, four 5s. Wait but maybe I miscounted. Let's count the number of 5s:

First 5: 1

Second 5: 2

Third term: d (not 5)

Fourth term: 5 (3)

Fifth term: d (not 5)

Sixth term: 5 (4)

Seventh term: d (not 5)

Ah! There we go. So 5 appears 4 times? Wait no, 1,2,4,6: that's four? Wait 1 (5), 2 (5), 4 (5), 6 (5): yes, four 5s. Wait but wait, the terms are 5,5,d,5,d,5,d. So 5 is multiplied 4 times? Wait 5×5×5×5 (four times) and d×d×d (three times). Wait, let's list the factors:

5: how many times? Let's count:

• First factor: 5 (1)

• Second factor: 5 (2)

• Fourth factor: 5 (3)

• Sixth factor: 5 (4)

Yes, 4 times. So the exponent for 5 is 4.

Step2: Count occurrences of d

Now count the number of times d appears. The terms with d are third, fifth, seventh: d, d, d. So that's 3 times. So the exponent for d is 3.

Answer:

Part A: 4

Part B: 3