Simplify the expression: (2^mn * 16^mn) / (8^(m-n) × 2^(m+n))

Question

ScanSolve AI Tutor · Accepted Answer

Step 1: Rewrite all bases as powers of 2. We know that 16 = 2⁴ and 8 = 2³. Substitute these into the expression: (2^(mn) × (2⁴)^(mn))/((2³)^(m−n) × 2^(m+n)) Step 2: Apply the exponent rule (a^b)^c = a^(bc). (2^(mn) × 2^(4 × mn))/(2^(3 × (m−n)) × 2^(m+n)) (2^(mn) × 2^(4mn))/(2^(3m−3n) × 2^(m+n)) Step 3: Apply the exponent rule a^b × a^c = a^(b+c) to combine terms in the numerator and denominator. Numerator: 2^(mn) × 2^(4mn) = 2^(mn + 4mn) = 2^(5mn) Denominator: 2^(3m−3n) × 2^(m+n) = 2^((3m−3n) + (m+n)) = 2^(3m−3n+m+n) = 2^(4m−2n) The expression now is: (2^(5mn))/(2^(4m−2n)) Step 4: Apply the exponent rule (a^b)/(a^c) = a^(b−c). 2^(5mn − (4m−2n)) Step 5: Simplify the exponent. 2^(5mn − 4m + 2n) The simplified expression is: 2^(5mn − 4m + 2n)

Simplify the expression: (2^mn * 16^mn) / (8^(m-n) * 2^(m+n))

Answer

Step-by-step solution

Rewrite all bases as powers of 2.

Apply the exponent rule $(a^b)^c = a^{bc}$ .

Apply the exponent rule $a^b \times a^c = a^{b+c}$ to combine terms in the numerator and denominator.

Apply the exponent rule $\frac{a^b}{a^c} = a^{b-c}$ .

Simplify the exponent.

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