Question

evaluate each expression.
a. $\frac{(1296m)^{\frac{3}{4}}}{1296^{\frac{1}{2}}}$
b. $(64^{-4})(64^{\frac{5}{3}})^2$
c. $(a^3b)^0(a^{\frac{2}{5}}b)^5$

Explanation:

Step1: Split numerator terms

$\frac{1296^{\frac{3}{4}}m^{\frac{3}{4}}}{1296^{\frac{1}{2}}}$

Step2: Subtract exponents of 1296

$1296^{\frac{3}{4}-\frac{1}{2}}m^{\frac{3}{4}} = 1296^{\frac{1}{4}}m^{\frac{3}{4}}$

Step3: Evaluate $1296^{\frac{1}{4}}$

$1296^{\frac{1}{4}} = 6$, so $6m^{\frac{3}{4}}$

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Step1: Simplify the exponent of 64

$(64^{-4})(64^{\frac{5}{3} \times 2}) = (64^{-4})(64^{\frac{10}{3}})$

Step2: Add exponents of 64

$64^{-4+\frac{10}{3}} = 64^{-\frac{2}{3}}$

Step3: Rewrite and evaluate $64^{-\frac{2}{3}}$

$64^{-\frac{2}{3}} = \frac{1}{(64^{\frac{1}{3}})^2} = \frac{1}{4^2} = \frac{1}{16}$

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Step1: Evaluate the zero exponent

$(a^3b)^0 = 1$, so $1 \times (a^{\frac{2}{5}}b)^5$

Step2: Distribute the exponent 5

$a^{\frac{2}{5} \times 5}b^{1 \times 5} = a^2b^5$

Answer:

a. $6m^{\frac{3}{4}}$
b. $\frac{1}{16}$
c. $a^2b^5$