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If x and y are positive integers and (3^{x})^{y} = 3^{5}, what is the value of 3^{x} \times 3^{y} ?

Explanation	Calculations	Help
$We recognize that the equation has unknowns in exponents, so we can equate the exponents on both sides using the theorem Prime Matching .$	$(3^{x})^{y} = 3^{5} 3^{x y} = 3^{5} x y = 5$	Theory & Tactics Method Card ALG7 Click to view full details
$We list the "positive integers" x and y that satisfy x y = 5 .$	$x y = 5 (x, y) = (1, 5) or (5, 1)$
$We then use the rule that multiplying powers with the same base adds the exponents, and then compute the resulting power step by step.$	$3^{x} \times 3^{y} = 3^{x + y} 3^{x + y} = 3^{6} 3^{6} = 3^{3} \times 3^{3} 3^{3} = 27 27 \times 27 = (20 + 7) \times (20 + 7) = 400 + 140 + 140 + 49 = 729$

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B

If x and y are positive integers and (3^{x})^{y} = 3^{5}, what is the value of 3^{x} \times 3^{y} ?

Explanation	Calculations	Help
$We recognize that the equation has unknowns in exponents, so we can equate the exponents on both sides using the theorem Prime Matching .$	$(3^{x})^{y} = 3^{5} 3^{x y} = 3^{5} x y = 5$	Theory & Tactics Method Card ALG7 Click to view full details
$We list the "positive integers" x and y that satisfy x y = 5 .$	$x y = 5 (x, y) = (1, 5) or (5, 1)$
$We then use the rule that multiplying powers with the same base adds the exponents, and then compute the resulting power step by step.$	$3^{x} \times 3^{y} = 3^{x + y} 3^{x + y} = 3^{6} 3^{6} = 3^{3} \times 3^{3} 3^{3} = 27 27 \times 27 = (20 + 7) \times (20 + 7) = 400 + 140 + 140 + 49 = 729$

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B