Multiply.
$\frac{3 b c}{2 c^4} \cdot \frac{8 c}{15 b}$
Simplify your answer as much as possible.
Answer (2)
Multiply the numerators and denominators: 2 c 4 3 b c ⋅ 15 b 8 c = 30 b c 4 24 b c 2 .
Simplify the numerical coefficients: 30 24 = 5 4 .
Cancel out the common factor b : b b = 1 .
Simplify the powers of c : c 4 c 2 = c 2 1 .
Combine the simplified terms: The final simplified expression is 5 c 2 4 .
Explanation
Understanding the Problem We are given the expression 2 c 4 3 b c ⋅ 15 b 8 c and we need to simplify it.
Multiplying Fractions First, multiply the numerators and the denominators: 2 c 4 3 b c ⋅ 15 b 8 c = ( 2 c 4 ) ( 15 b ) ( 3 b c ) ( 8 c )
Simplifying Numerator and Denominator Now, simplify the numerator and the denominator separately.
Numerator: ( 3 b c ) ( 8 c ) = 3 ⋅ 8 ⋅ b ⋅ c ⋅ c = 24 b c 2
Denominator: ( 2 c 4 ) ( 15 b ) = 2 ⋅ 15 ⋅ b ⋅ c 4 = 30 b c 4
So, the expression becomes: 30 b c 4 24 b c 2
Canceling Common Factors Next, we simplify the fraction by canceling out common factors. First, we simplify the numerical coefficients: 30 24 = 6 ⋅ 5 6 ⋅ 4 = 5 4 Then, we cancel out the common factor b :
b b = 1 Finally, we simplify the powers of c :
c 4 c 2 = c 4 − 2 1 = c 2 1
Final Simplification Combining all the simplified terms, we get: 5 4 ⋅ c 2 1 = 5 c 2 4
Final Answer Therefore, the simplified expression is 5 c 2 4 .
Examples
When calculating the area of a complex shape, you might end up with an expression like the one we simplified. Simplifying such expressions helps in getting a more manageable formula for the area, making it easier to compute for different values. For instance, if c represents a variable dimension of the shape, the simplified expression allows for quick area calculations as c changes. This is also useful in physics when dealing with rates and ratios, where simplifying expressions can lead to clearer understanding and easier computations.
To simplify the expression 2 c 4 3 b c ⋅ 15 b 8 c , first multiply the fractions to get 30 b c 4 24 b c 2 . Then simplify by canceling common factors, resulting in the final answer 5 c 2 4 .
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