Question

How do you simplify $ {100^{\dfrac{{ - 3}}{2}}} $ ?

Answer 1

Hint: We will start by using the following exponent rule:
 $ {({x^a})^b} = {x^{a \times b}} $ . Mention all the terms. Then reduce the terms until they cannot be reduced any further. We will also use the rule $ {x^a} = \dfrac{1}{{{x^{ - a}}}} $ to simplify the terms further.

Complete step-by-step answer:
We will start off by applying the exponent rule given by, $ {({x^a})^b} = {x^{a \times b}} $ .
 \[
   = {100^{\dfrac{{ - 3}}{2}}} \\
   = {100^{\dfrac{1}{2} \times - 3}} \\
   = {\left( {{{100}^{\dfrac{1}{2}}}} \right)^{ - 3}} \;
 \]
Now we will simplify the terms which are within the parenthesis.
 \[
   = {\left( {{{100}^{\dfrac{1}{2}}}} \right)^{ - 3}} \\
   = {\left( {{{\left( {{{10}^2}} \right)}^{\dfrac{1}{2}}}} \right)^{ - 3}} \\
   = {10^{ - 3}} \;
 \]
Now here, we will apply the following rule, to reduce the terms further.
 $ {x^a} = \dfrac{1}{{{x^{ - a}}}} $
 $
   = {10^{ - 3}} \\
   = \dfrac{1}{{{{10}^3}}} \\
   = \dfrac{1}{{10 \times 10 \times 10}} \\
   = \dfrac{1}{{1000}} \\
   = 0.001 \;
  $
So, the correct answer is “0.001”.

Note: The power rule tells us that to raise a power to a power, just multiply the exponents. The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. If any negative number is raised to a negative power equals its reciprocal raised to the opposite power. Any number raised to a power one equals itself. In the exponent product rule, when multiplying two powers that have the same base, you can add the exponents. A negative exponent means divides, because the opposite of multiplying is dividing. A fractional exponent means to take the $ {n^{th}} $ root.