Simplify the expression 4a^(-3)b^(3)*2a^(2)b^(-1)*4a^(2)b^(2)

Apply Exponent Properties: To simplify the expression, we will use the properties of exponents, which state that when multiplying powers with the same base, we add the exponents. We will apply this rule to both the 'a' terms and the 'b' terms separately. Calculation: For the 'a' terms: a^(-3) * a^(2) * a^(2) = a^(-3 + 2 + 2) = a^(1) For the 'b' terms: b^(3) * b^(-1) * b^(2) = b^(3 - 1 + 2) = b^(4)Calculate 'a' Terms: Next, we multiply the coefficients together. The coefficients are 4, 2, and 4. Calculation: 4 * 2 * 4 = 32Calculate 'b' Terms: Now we combine the results from the previous steps to form the simplified expression. Calculation: 32 * a^(1) * b^(4) = 32a * b^(4)

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Q. Simplify the expression :

4 a^{-3} b^{3} \cdot 2 a^{2} b^{-1} \cdot 4 a^{2} b^{2}

Apply Exponent Properties: To simplify the expression, we will use the properties of exponents, which state that when multiplying powers with the same base, we add the exponents. We will apply this rule to both the ' $a$ ' terms and the ' $b$ ' terms separately. $\newline$ Calculation: $\newline$ For the ' $a$ ' terms: $a^{-3} \times a^{2} \times a^{2} = a^{-3 + 2 + 2} = a^{1}$ $\newline$ For the ' $b$ ' terms: $b^{3} \times b^{-1} \times b^{2} = b^{3 - 1 + 2} = b^{4}$
Calculate 'a' Terms: Next, we multiply the coefficients together. The coefficients are $4$ , $2$ , and $4$ . $\newline$ Calculation: $4 \times 2 \times 4 = 32$
Calculate 'b' Terms: Now we combine the results from the previous steps to form the simplified expression. $\newline$ Calculation: $32 \times a^{1} \times b^{4} = 32a \times b^{4}$