Answer:
(a) m = 10
(b) n = 160
Step-by-step explanation:
To rewrite the expression in the manner indicated, write an equation that sets the given expression equal to the desired expression. Then solve for the variable value. The distributive property and the usual rules of exponents apply.
The rules of exponents you can use here are ...
(a^b)(a^c) = a^(b+c)
(a^b)^c = a^(bc)
a^0 = 1
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(a) 2^101 +2^103 = m×2^100
2^(101-100) +2^(103-100) = m × 2^(100-100) . . . . . . . multiply by 2^(-100)
2^1 +2^3 = m
2 + 8 = m
m = 10
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(b) 2^101 +2^103 = n×4^48
2^101 +2^103 = n×(2^2)^48 . . . . . substitute 4 = 2^2
2^101 +2^103 = n×2^96 . . . . . . . . simplify
2^5 +2^7 = n . . . . . . . . . . . . . . . . . as above, multiply by 2^(-96)
32 +128 = n
n = 160
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<em>Additional comment</em>
The rules of exponents tell you that ...
(m × 2^100) × 2^-100 = m × 2^(100 -100) = m × 2^0 = m
This is effectively the same as dividing by 2^100. You're doing the same thing you would do to solve any linear equation: divide by the coefficient of the variable.
