(a) [tex]\frac{54^2 \times 12^3 \times 64^2\left(3^2 \times 4^3 \times 5^2\right)^3}{\left(3^2 \times 15 \times 20^3\right)^4}[/tex]
Answer (2)
Express all numbers in the expression in terms of their prime factorizations.
Simplify the expression by applying exponent rules.
Cancel out common factors in the numerator and denominator.
Calculate the final numerical value: 9765625 442368
Explanation
Prime Factorization We are asked to simplify the expression ( 3 2 × 15 × 2 0 3 ) 4 5 4 2 × 1 2 3 × 6 4 2 ( 3 2 × 4 3 × 5 2 ) 3 To do this, we will express each number as a product of its prime factors.
Prime Factors of Bases First, let's find the prime factorization of each base: 54 = 2 × 3 3 12 = 2 2 × 3 64 = 2 6 3 2 = 3 2 4 = 2 2 5 2 = 5 2 15 = 3 × 5 20 = 2 2 × 5
Substituting and Simplifying Now, substitute these factorizations into the expression: ( 3 2 × ( 3 × 5 ) × ( 2 2 × 5 ) 3 ) 4 ( 2 × 3 3 ) 2 × ( 2 2 × 3 ) 3 × ( 2 6 ) 2 ( 3 2 × ( 2 2 ) 3 × 5 2 ) 3 Simplify the exponents: ( 3 2 × 3 × 5 × 2 6 × 5 3 ) 4 ( 2 2 × 3 6 ) × ( 2 6 × 3 3 ) × ( 2 12 ) ( 3 2 × 2 6 × 5 2 ) 3 ( 3 3 × 5 × 2 6 × 5 3 ) 4 ( 2 2 × 3 6 ) × ( 2 6 × 3 3 ) × ( 2 12 ) ( 3 6 × 2 18 × 5 6 ) ( 3 3 × 5 4 × 2 6 ) 4 2 2 × 3 6 × 2 6 × 3 3 × 2 12 × 3 6 × 2 18 × 5 6 Combine like terms in the numerator: 3 3 × 4 × 5 4 × 4 × 2 6 × 4 2 2 + 6 + 12 + 18 × 3 6 + 3 + 6 × 5 6 3 12 × 5 16 × 2 24 2 38 × 3 15 × 5 6
Cancelling Common Factors Now, simplify by cancelling out common factors: 2 38 − 24 × 3 15 − 12 × 5 6 − 16 2 14 × 3 3 × 5 − 10 5 10 2 14 × 3 3 9765625 16384 × 27 = 9765625 442368
Final Result The simplified expression is 9765625 442368 = 0.0452984832 .
Examples
When dealing with scaling recipes or adjusting ingredient quantities, simplifying complex fractions becomes essential. For instance, if a recipe calls for multiplying ingredient amounts by a complex fraction, simplifying this fraction beforehand ensures accurate measurements and consistent results. This skill is also valuable in fields like chemistry, where reactions involve precise molar ratios represented by fractions, and in finance, where investment returns or ratios are often expressed as fractions that need simplification for clear interpretation.
To simplify the expression ( 3 2 × 15 × 2 0 3 ) 4 5 4 2 × 1 2 3 × 6 4 2 ( 3 2 × 4 3 × 5 2 ) 3 , we used prime factorization, combined like terms, and cancelled common factors to arrive at 9765625 442368 .
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