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If y and m are integers and (2 0^{y}) (4 0^{y - 1}) = (2^{m}) (5^{3}), what is the value of m ?

Explanation	Calculations	Help
$We see unknowns in exponents, so we use Prime Matching and start by expressing each side of the equation in prime factors$	$2 0^{y} = (2^{2} \times 5)^{y} = 2^{2 y} \times 5^{y} 4 0^{y - 1} = (2^{3} \times 5)^{y - 1} = 2^{3 y - 3} \times 5^{y - 1}$	Theory & Tactics Method Card ALG7 Click to view full details
$We combine like bases on the left side$	$2 0^{y} \times 4 0^{y - 1} = 2^{2 y} \times 5^{y} \times 2^{3 y - 3} \times 5^{y - 1} = 2^{5 y - 3} \times 5^{2 y - 1}$
$We match the exponent of 5 from both sides$	$2^{5 y - 3} \times 5^{2 y - 1} = 2^{m} \times 5^{3} 5 -exponent: 2 y - 1 = 3 2 y = 4 y = 2$	Theory & Tactics Method Card ALG7 Click to view full details
$We match the exponent of 2 and substitute y = 2$	$2 -exponent: 5 y - 3 = m 5 (2) - 3 = m 10 - 3 = m m = 7$	Theory & Tactics Method Card ALG7 Click to view full details

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C

If y and m are integers and (2 0^{y}) (4 0^{y - 1}) = (2^{m}) (5^{3}), what is the value of m ?

Explanation	Calculations	Help
$We see unknowns in exponents, so we use Prime Matching and start by expressing each side of the equation in prime factors$	$2 0^{y} = (2^{2} \times 5)^{y} = 2^{2 y} \times 5^{y} 4 0^{y - 1} = (2^{3} \times 5)^{y - 1} = 2^{3 y - 3} \times 5^{y - 1}$	Theory & Tactics Method Card ALG7 Click to view full details
$We combine like bases on the left side$	$2 0^{y} \times 4 0^{y - 1} = 2^{2 y} \times 5^{y} \times 2^{3 y - 3} \times 5^{y - 1} = 2^{5 y - 3} \times 5^{2 y - 1}$
$We match the exponent of 5 from both sides$	$2^{5 y - 3} \times 5^{2 y - 1} = 2^{m} \times 5^{3} 5 -exponent: 2 y - 1 = 3 2 y = 4 y = 2$	Theory & Tactics Method Card ALG7 Click to view full details
$We match the exponent of 2 and substitute y = 2$	$2 -exponent: 5 y - 3 = m 5 (2) - 3 = m 10 - 3 = m m = 7$	Theory & Tactics Method Card ALG7 Click to view full details

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C