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((1)/(9))^(-(3)/(2))*((1)/(8))^((2)/(3)) What is the value of the given expression?

(19)−32⋅(18)23 \left(\frac{1}{9}\right)^{-\frac{3}{2}} \cdot\left(\frac{1}{8}\right)^{\frac{2}{3}} \newlineWhat is the value of the given expression?

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Full solution

Q. (19)−32⋅(18)23 \left(\frac{1}{9}\right)^{-\frac{3}{2}} \cdot\left(\frac{1}{8}\right)^{\frac{2}{3}} \newlineWhat is the value of the given expression?
  1. Understand and Simplify Exponents: Understand the given expression and simplify the exponents.\newlineThe given expression is (19)−32×(18)23\left(\frac{1}{9}\right)^{-\frac{3}{2}}\times\left(\frac{1}{8}\right)^{\frac{2}{3}}. We need to simplify each part of the expression separately.
  2. Simplify ((1)/(9))(−(3)/(2))((1)/(9))^(-(3)/(2)): Simplify the first part of the expression ((1)/(9))(−(3)/(2))((1)/(9))^(-(3)/(2)). The negative exponent means we take the reciprocal of the base and then apply the positive exponent. So, ((1)/(9))(−(3)/(2))((1)/(9))^(-(3)/(2)) becomes (9)(3/2)(9)^{(3/2)}.
  3. Calculate (9)(3/2)(9)^{(3/2)}: Calculate (9)(3/2)(9)^{(3/2)}.\newlineSince 99 is a perfect square (32)(3^2), we can rewrite (9)(3/2)(9)^{(3/2)} as (32)(3/2)(3^2)^{(3/2)}. When we raise a power to a power, we multiply the exponents, so (32)(3/2)=3(2∗(3/2))=33(3^2)^{(3/2)} = 3^{(2*(3/2))} = 3^3.\newlineNow calculate 33=3×3×3=273^3 = 3 \times 3 \times 3 = 27.
  4. Simplify ((1)/(8))((2)/(3))((1)/(8))^((2)/(3)): Simplify the second part of the expression ((1)/(8))((2)/(3))((1)/(8))^((2)/(3)). The exponent (2/3)(2/3) means we take the cube root of the base and then square it. So, ((1)/(8))((2)/(3))((1)/(8))^((2)/(3)) becomes (1/2)2(1/2)^2.
  5. Calculate (1/2)2(1/2)^2: Calculate (1/2)2(1/2)^2.\newlineSquare the fraction (1/2)(1/2) to get (1/2)2=1/4(1/2)^2 = 1/4.
  6. Multiply Results: Multiply the results from Step 33 and Step 55.\newlineNow we multiply 2727 (from Step 33) by 14\frac{1}{4} (from Step 55) to get the final value of the expression.\newline27×(14)=274=6.7527 \times \left(\frac{1}{4}\right) = \frac{27}{4} = 6.75.

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