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If x and y are positive integers and (3^{x})^{y} = 3^{7}, what is the value of 3^{x} im es 3^{y} ?

Explanation	Calculations	Help
$We see that the equation has unknowns in the exponents, so we apply Prime Matching .$	$(3^{x})^{y} = 3^{7} 3^{x y} = 3^{7} x y = 7$	Theory & Tactics Method Card ALG7 Click to view full details
$We list all positive integer pairs (x, y) such that x y = 7 .$	$x y = 7 (x, y) = (1, 7) or (7, 1)$
$We then write the product as a single power of three and evaluate it by breaking it down into smaller exponents.$	$3^{x} \times 3^{y} = 3^{x + y} x + y = 1 + 7 = 8 3^{8} = 3^{4} \times 3^{4} 3^{4} = 3^{2} \times 3^{2} = 9 \times 9 = 81 3^{8} = 81 \times 81 81 \times 81 = (80 + 1) \times (80 + 1) = 6400 + 80 + 80 + 1 = 6561$

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C

If x and y are positive integers and (3^{x})^{y} = 3^{7}, what is the value of 3^{x} im es 3^{y} ?

Explanation	Calculations	Help
$We see that the equation has unknowns in the exponents, so we apply Prime Matching .$	$(3^{x})^{y} = 3^{7} 3^{x y} = 3^{7} x y = 7$	Theory & Tactics Method Card ALG7 Click to view full details
$We list all positive integer pairs (x, y) such that x y = 7 .$	$x y = 7 (x, y) = (1, 7) or (7, 1)$
$We then write the product as a single power of three and evaluate it by breaking it down into smaller exponents.$	$3^{x} \times 3^{y} = 3^{x + y} x + y = 1 + 7 = 8 3^{8} = 3^{4} \times 3^{4} 3^{4} = 3^{2} \times 3^{2} = 9 \times 9 = 81 3^{8} = 81 \times 81 81 \times 81 = (80 + 1) \times (80 + 1) = 6400 + 80 + 80 + 1 = 6561$

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C