EXPRESS THE FOLLOWING IN SIMPLEST EXPONENTIAL FORM

Express in simplest form with a prime number base :

Problem 1 :

4

Solution :

Decompose 4 into prime factors.

4 = 2×2

2 is repeated 2 times.

= 2² 

Problem 2 :

16

Solution :

Decompose 16 into prime factors.

16 = 2×2×2×2

2 is repeated 4 times.

= 2⁴ 

Problem 3 :

27

Solution :

Decompose 27 into prime factors.

27 = 3×3×3

3 is repeated 3 times.

= 3³ 

Problem 4 :

4²

Solution :

4² = (2×2)²

2 is repeated 2 times.

= (2²)²

By using power rule (a^m)ⁿ = a^mn, we get

= 2⁴  

Problem 5 :

25²

Solution :

25² = (5×5)²

5 is repeated 2 times.

= (5²)²

By using power rule (a^m)ⁿ = a^mn, we get

= 5⁴ 

Problem 6 :

2^t × 8

Solution :

Decompose 8 into prime factors.

8 = 2×2×2

= 2^t × 8

= 2^t×2×2×2

2 is repeated 3 times.

= 2^t × 2³

By using product rule a^m × aⁿ = a^m+n, we get

= 2^{t + 3}

Problem 7 :

3^a ÷ 3

Solution :

= 3^a ÷ 3

= 3^a ÷ 3¹

By using quotient rule a^m ÷ aⁿ = a^m-n, we get

= 3^{a – 1} 

Problem 8 :

2ⁿ × 4ⁿ

Solution :

= 2ⁿ × 4ⁿ

= 2ⁿ×(2×2)ⁿ

= 2ⁿ × (2²)ⁿ

= (2)ⁿ × (2)²ⁿ

By using product rule a^m × aⁿ = a^m+n, we get

= 2^{n+ 2n}

= 2³ⁿ 

Problem 9 :

9/(3^x)

Solution :

= 9/(3^x)

= (3²)/(3^x)

By using quotient rule a^m/aⁿ = a^m-n, we get

= 3^{2 – x}

Problem 10 :

5ⁿ⁺²/5^n–2

Solution :

= 5ⁿ⁺²/5^n–2

By using quotient rule a^m/aⁿ = a^m-n, we get

= 5^{(n+2)–(n–2)}

= 5^n+2-n+2

= 5²⁺²

= 5⁴

Problem 11 :

(3⁴)^a+1

Solution :

= (3⁴)^a+1

By using power rule (a^m)ⁿ = a^mn, we get

= 3^{(4a + 4)} 

Problem 12 :

3^x × 3^4-x

Solution :

= 3^x × 3^4–x

By using product rule a^m × aⁿ = a^m+n, we get

= (3)^x+4–x

= 3⁴ 

Problem 13 :

(4^a)/2^b

Solution :

= (4^a)/2^b

= (2 × 2)^a/2^b

= (2²)^a/2^b

= 2^2a/2^b

By using quotient rule a^m/aⁿ = a^m-n, we get

= 2^{(2a – b)} 

Problem 14 :

8^x/16^y

Solution :

8^x/16^y = (2 × 2 × 2)^x/(2 × 2 × 2 × 2)^y

= (2³)^x/(2⁴)^y

 = 2^3x/2^4y

By using quotient rule a^m/aⁿ = a^m-n, we get

= 2^{3x – 4y} 

Problem 15 :

5^1+x /5^x-1

Solution :

= 5^1+x/5^x–1 

By using quotient rule a^m/aⁿ = a^m-n, we get

= 5^1+x-x+1

= 5²

Problem 16 :

(3^a × 9^a)/27^a+2

Solution :

= (3^a × (3²)^a)/(3³) ^{a + 2}

= (3^a × 3^2a)/3^3a+6

= 3^a+2a/3^3a+6

= 3^a+2a-3a-6

= 3^a-a-6

= 3^-6

= 1/3⁶

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