Solved on Dec 14, 2023

Find the prime factorization of 72 using exponential notation: $2^{3} \cdot 3^{2}$

STEP 1

Assumptions
1. We need to find the prime factorization of 72.
2. Prime factorization is expressing a number as the product of its prime factors.
3. Exponential notation means using exponents to show how many times a prime number is used as a factor.

STEP 2

Begin by writing 72 as a product of any two factors.
$72 = 8 \times 9$

STEP 3

Notice that neither 8 nor 9 is a prime number, so we need to further factorize both numbers.

STEP 4

Factorize 8 into prime factors.
$8 = 2 \times 4$

STEP 5

Further factorize 4, which is a factor of 8.
$4 = 2 \times 2$

STEP 6

Combine the factors of 8 into prime factors using exponential notation.
$8 = 2 \times 2 \times 2 = 2^{3}$

STEP 7

Now, factorize 9 into prime factors.
$9 = 3 \times 3$

STEP 8

Write 9 as a prime factor using exponential notation.
$9 = 3 \times 3 = 3^{2}$

STEP 9

Combine the prime factors of 8 and 9 to express 72 as a product of prime factors.
$72 = 2^{3} \times 3^{2}$

STEP 10

Compare the result with the given options.
The correct prime factorization of 72 using exponential notation is $2^{3} \cdot 3^{2}$ .

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Find the prime factorization of 72 using exponential notation: $2^{3} \cdot 3^{2}$

STEP 1

Assumptions
1. We need to find the prime factorization of 72.
2. Prime factorization is expressing a number as the product of its prime factors.
3. Exponential notation means using exponents to show how many times a prime number is used as a factor.

STEP 2

Begin by writing 72 as a product of any two factors.
$72 = 8 \times 9$

STEP 3

Notice that neither 8 nor 9 is a prime number, so we need to further factorize both numbers.

STEP 4

Factorize 8 into prime factors.
$8 = 2 \times 4$

STEP 5

Further factorize 4, which is a factor of 8.
$4 = 2 \times 2$

STEP 6

Combine the factors of 8 into prime factors using exponential notation.
$8 = 2 \times 2 \times 2 = 2^{3}$

STEP 7

Now, factorize 9 into prime factors.
$9 = 3 \times 3$

STEP 8

Write 9 as a prime factor using exponential notation.
$9 = 3 \times 3 = 3^{2}$

STEP 9

Combine the prime factors of 8 and 9 to express 72 as a product of prime factors.
$72 = 2^{3} \times 3^{2}$

STEP 10

Compare the result with the given options.
The correct prime factorization of 72 using exponential notation is $2^{3} \cdot 3^{2}$ .

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Find the prime factorization of 72 using exponential notation: 23⋅322^{3} \cdot 3^{2}23⋅32

STEP 1

Assumptions1. We need to find the prime factorization of 72.2. Prime factorization is expressing a number as the product of its prime factors.3. Exponential notation means using exponents to show how many times a prime number is used as a factor.

STEP 2

Begin by writing 72 as a product of any two factors.72=8×972 = 8 \times 972=8×9

STEP 3

Notice that neither 8 nor 9 is a prime number, so we need to further factorize both numbers.

STEP 4

Factorize 8 into prime factors.8=2×48 = 2 \times 48=2×4

STEP 5

Further factorize 4, which is a factor of 8.4=2×24 = 2 \times 24=2×2

STEP 6

Combine the factors of 8 into prime factors using exponential notation.8=2×2×2=238 = 2 \times 2 \times 2 = 2^{3}8=2×2×2=23

STEP 7

Now, factorize 9 into prime factors.9=3×39 = 3 \times 39=3×3

STEP 8

Write 9 as a prime factor using exponential notation.9=3×3=329 = 3 \times 3 = 3^{2}9=3×3=32

STEP 9

Combine the prime factors of 8 and 9 to express 72 as a product of prime factors.72=23×3272 = 2^{3} \times 3^{2}72=23×32

STEP 10

Compare the result with the given options.The correct prime factorization of 72 using exponential notation is 23⋅322^{3} \cdot 3^{2}23⋅32.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the prime factorization of 72 using exponential notation: $2^{3} \cdot 3^{2}$

Assumptions
1. We need to find the prime factorization of 72.
2. Prime factorization is expressing a number as the product of its prime factors.
3. Exponential notation means using exponents to show how many times a prime number is used as a factor.

Begin by writing 72 as a product of any two factors.
$72 = 8 \times 9$

Factorize 8 into prime factors.
$8 = 2 \times 4$

Further factorize 4, which is a factor of 8.
$4 = 2 \times 2$

Combine the factors of 8 into prime factors using exponential notation.
$8 = 2 \times 2 \times 2 = 2^{3}$

Now, factorize 9 into prime factors.
$9 = 3 \times 3$

Write 9 as a prime factor using exponential notation.
$9 = 3 \times 3 = 3^{2}$

Combine the prime factors of 8 and 9 to express 72 as a product of prime factors.
$72 = 2^{3} \times 3^{2}$

Compare the result with the given options.
The correct prime factorization of 72 using exponential notation is $2^{3} \cdot 3^{2}$ .