Question

plz Solve this :-


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Answer 1

Step-by-step explanation:

I hope u understood

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Answer 2

Question: To simplify:

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$\dashrightarrow \begin{array}{c}\Large \text{\sf (25)}^{\sf \dfrac{1}{2}} \times \Large \text{\sf (243)}^{\sf \dfrac{3}{5}}\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\\Large \text{\sf (16)}^{\sf \dfrac{5}{4}} \times \Large \text{\sf (8)}^{\sf \dfrac{4}{3}}\end{array}$

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Solution:

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We know that;

• Prime factorization of 25 = 5 × 5

• Prime factorization of 243 = 3 × 3 × 3 × 3 × 3

• Prime factorization of 16 = 2 × 2 × 2 × 2

• Prime factorization of 8 = 2 × 2 × 2

On substituting them we get;

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$\dashrightarrow \begin{array}{c}\Large \text{\sf (5 x 5)}^{\sf \dfrac{1}{2}} \times \Large \text{\sf (3 x 3 x 3 x 3 x 3)}^{\sf \dfrac{3}{5}}\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\\Large \text{\sf (2 x 2 x 2 x 2)}^{\sf \dfrac{5}{4}} \times \Large \text{\sf (2 x 2 x 2)}^{\sf \dfrac{4}{3}}\end{array}$

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$\dashrightarrow \begin{array}{c}\Large \text{\sf (5}^{\sf 2}\Large \text{\sf )}^{\sf \dfrac{1}{2}} \times \Large \text{\sf (3}^{\sf 5} \Large \text{\sf)}^{\sf \dfrac{3}{5}}\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\\Large \text{\sf (2}^{\sf4} \Large \text{\sf )}^{\sf \dfrac{5}{4}} \times \Large \text{\sf (2}^{\sf 3}\Large \text{\sf)}^{\sf \dfrac{4}{3}}\end{array}$

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$\dashrightarrow \begin{array}{c}\Large \text{\sf (5)}^{\sf 2 \times \dfrac{1}{2}} \times \Large \text{\sf (3)}^{\sf 5 \times \dfrac{3}{5}}\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\\Large \text{\sf (2)}^{\sf 4 \times \dfrac{5}{4}} \times \Large \text{\sf (2)}^{\sf 3 \times \dfrac{4}{3}}\end{array}$

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$\dashrightarrow \begin{array}{c}\Large \text{\sf (5)}^{\sf \left\{\dfrac{2}{2}\right\}} \times \Large \text{\sf (3)}^{\sf \left\{\dfrac{5 \times3}{5}\right\}}\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\\Large \text{\sf (2)}^{\sf \left\{\dfrac{4 \times 5}{4}\right\}} \times \Large \text{\sf (2)}^{\sf \left\{\dfrac{3 \times 4}{3}\right\}}\end{array}$

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$\dashrightarrow \begin{array}{c}\Large \text{\sf (5)}^{\sf \left\{\dfrac{\not{2}}{\not{2}}\right\}} \times \Large \text{\sf (3)}^{\sf \left\{\dfrac{\not{5} \times 3}{\not{5}}\right\}}\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\\Large \text{\sf (2)}^{\sf \left\{\dfrac{\not{4} \times 5}{\not{4}}\right\}} \times \Large \text{\sf (2)}^{\sf \left\{\dfrac{\not{3} \times 4}{\not{3}}\right\}}\end{array}$

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$\dashrightarrow \begin{array}{c}\Large \text{\sf (5)}^{\sf 1} \times \Large \text{\sf (3)}^{\sf 3}\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\\Large \text{\sf (2)}^{\sf 5} \times \Large \text{\sf (2)}^{\sf 4}\end{array}$

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Using the exponent law mᵃ × mᵇ = $\sf m^{\sf(a + b)}$  in the denominator we get;

• [Where m = 2, a = 5, and b = 4]

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$\dashrightarrow \begin{array}{c}\Large \text{\sf 5} \times \Large \text{\sf (3 x 3 x 3)}\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\\Large \text{\sf (2)}^{\sf (5 + 4)}\end{array}$

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$\dashrightarrow \ \dfrac{\large \text{\sf 5} \times \large \text{\sf27}}{\large \text{\sf 2}^{\sf 9}}$

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$\dashrightarrow \ \dfrac{\large \text{\sf 135}}{\large \text{\sf512}}$

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Hence simplified.