((2^(3))^(4)*(2^(34)))/((2^(5))^(8))

Identify and Apply Power Rule: Identify the given expression and apply the power of a power rule which states that (a^b)^c = a^(b*c). The expression is ((2^(3))^(4)*(2^(34)))/((2^(5))^(8)). First, simplify (2^(3))^(4) by multiplying the exponents. (2^(3))^(4) = 2^(3*4) = 2^(12).Simplify Exponent (2^3)^4: Now, simplify the denominator (2^(5))^(8) by multiplying the exponents. (2^(5))^(8) = 2^(5*8) = 2^(40).Simplify Exponent (2^5)^8: Combine the simplified parts of the expression. The expression now is (2^(12)*2^(34))/2^(40).Combine Simplified Parts: Apply the product of powers rule which states that a^m * a^n = a^(m+n) to combine the powers of 2 in the numerator. 2^(12)*2^(34) = 2^(12+34) = 2^(46).Apply Product of Powers Rule: Now, apply the quotient of powers rule which states that a^m / a^n = a^(m-n) to divide the powers of 2. (2^(46))/2^(40) = 2^(46-40) = 2^(6).Apply Quotient of Powers Rule: Finally, simplify 2^(6). 2^(6) = 2*2*2*2*2*2 = 64.

Home

Math Problems

Precalculus

Operations with rational exponents

Full solution

\frac{\left(2^{3}\right)^{4} \cdot\left(2^{34}\right)}{\left(2^{5}\right)^{8}}

Identify and Apply Power Rule: Identify the given expression and apply the power of a power rule which states that $(a^b)^c = a^{(b*c)}$ . $\newline$ The expression is $\left((2^{3})^{4}\cdot(2^{34})\right)/\left((2^{5})^{8}\right)$ . $\newline$ First, simplify $(2^{3})^{4}$ by multiplying the exponents. $\newline$ $(2^{3})^{4} = 2^{3\cdot4} = 2^{12}$ .
Simplify Exponent $(2^3)^4$ : Now, simplify the denominator $(2^{5})^{8}$ by multiplying the exponents. $\newline$ $(2^{5})^{8} = 2^{5*8} = 2^{40}.$
Simplify Exponent $(2^5)^8$ : Combine the simplified parts of the expression. $\newline$ The expression now is $\frac{2^{12} \cdot 2^{34}}{2^{40}}$ .
Combine Simplified Parts: Apply the product of powers rule which states that $a^m \times a^n = a^{(m+n)}$ to combine the powers of $2$ in the numerator. $\newline$ $2^{12}\times2^{34} = 2^{12+34} = 2^{46}.$
Apply Product of Powers Rule: Now, apply the quotient of powers rule which states that $a^m / a^n = a^{(m-n)}$ to divide the powers of $2$ . $\newline$ $(2^{46})/2^{40} = 2^{(46-40)} = 2^{6}$ .
Apply Quotient of Powers Rule: Finally, simplify $2^{6}$ . $2^{6} = 2\times2\times2\times2\times2\times2 = 64$ .