Evaluate (a) (8^(1/3) * 0.333... - 30^(-1)) / (3^(1/2) * 3^(1.5/3) - 1/15) and (b) (2.1333... / (5^(-3) + 1/3))^(-3/2), giving exact fractions.

Step-by-step solution

1. 1

   Convert every repeating decimal to a fraction before touching the exponents. For a purely repeating decimal, let $x=0.333\ldots$; then $10x=3.333\ldots$, and subtracting gives $9x=3$, so $x=\frac39=\frac13$. Both names are the same number — $\frac39$ is simply not in lowest terms.

   For $2.133\ldots$ the repeating block starts after one fixed digit, so use two multipliers: with $y=0.1333\ldots$, $100y-10y=13.33\ldots-1.33\ldots=12$, giving $y=\frac{12}{90}=\frac{2}{15}$ and

   $2.133\ldots=2+\frac{2}{15}=\frac{32}{15}$

   A warning about a very common trap: $0.0333\ldots=\frac{1}{30}$, not $\frac{3}{9}$. Shifting the block one place right divides the value by $10$.

2. 2

   Simplify the rational exponents in (a). $8^{1/3}=\sqrt[3]{8}=2$, and since $1{,}5=\frac32$ we get $\frac{1{,}5}{3}=\frac12$, so

   $3^{\frac12}\times3^{\frac12}=3^{\frac12+\frac12}=3^{1}=3$

   Adding exponents on a common base is what makes this denominator collapse to a whole number.

3. 3

   Evaluate the numerator and denominator of (a) separately.

   $\text{numerator}=2\cdot\frac13-\frac{1}{30}=\frac{20}{30}-\frac{1}{30}=\frac{19}{30}$

   $\text{denominator}=3-\frac{1}{15}=\frac{45}{15}-\frac{1}{15}=\frac{44}{15}$

   $\frac{19/30}{44/15}=\frac{19}{30}\times\frac{15}{44}=\frac{19}{2\cdot44}=\frac{19}{88}$

4. 4

   Build the inner fraction of (b). With $5^{-3}=\frac{1}{125}$,

   $5^{-3}+\frac13=\frac{3}{375}+\frac{125}{375}=\frac{128}{375}$

   $\frac{32/15}{128/375}=\frac{32}{15}\times\frac{375}{128}=\frac{32\times25}{128}=\frac{25}{4}$

   The numbers were chosen so this lands on a perfect square over a perfect square, which is the hint for the next step.

5. 5

   Apply the negative fractional exponent. A negative exponent flips the fraction and the exponent $\frac32$ means "square root, then cube":

   $\left(\frac{25}{4}\right)^{-\frac32}=\left(\frac{4}{25}\right)^{\frac32}=\left[\left(\frac{2}{5}\right)^{2}\right]^{\frac32}=\left(\frac{2}{5}\right)^{3}=\frac{8}{125}$

6. 6

   Check both answers as decimals. $\frac{19}{88}=0.2159\ldots$, and directly $\frac{0.6333}{2.9333}=0.2159$. For (b), $\frac{8}{125}=0.064$, and $6.25^{-1.5}=0.064$. Both match.