Question

Express as a power of a rational number with positive exponent: \[{\left[ {{{\left( {\dfrac{4}{5}} \right)}^{ - 2}}} \right]^4}\]. If the answer is \[{\left( {\dfrac{5}{4}} \right)^m}\], find the value of \[m\].

Answer 1

Hint: Here, we need to express the given expression as a rational number with positive exponent, and find the value of \[m\]. We will use the rules of exponents to simplify the given expression to a rational number with a positive exponent. Then, we will equate it to \[{\left( {\dfrac{5}{4}} \right)^m}\], and compare them to find the value of \[m\].

Formula Used: We will use the rules of exponents:
If a number raised to an exponent is again raised to an exponent, the new power is the product of the exponents. This can be written as \[{\left( {{a^b}} \right)^c} = {a^{b \times c}}\].

Complete step-by-step answer:
We will use the rule of exponents to express the expression \[{\left[ {{{\left( {\dfrac{4}{5}} \right)}^{ - 2}}} \right]^4}\] as a rational number with positive exponent.
We know that \[{\left( {{a^b}} \right)^c} = {a^{b \times c}}\].
We will use this rule to simplify the given expression.
Substituting \[a = \dfrac{4}{5}\], \[b = - 2\], and \[c = 4\] in the equation, we get
\[ \Rightarrow {\left[ {{{\left( {\dfrac{4}{5}} \right)}^{ - 2}}} \right]^4} = {\left( {\dfrac{4}{5}} \right)^{ - 2 \times 4}}\]
Multiplying the numbers in the exponent, we get
\[ \Rightarrow {\left[ {{{\left( {\dfrac{4}{5}} \right)}^{ - 2}}} \right]^4} = {\left( {\dfrac{4}{5}} \right)^{ - 8}}\]
Now, we will rewrite the exponent as a positive exponent.
Suppose that a number \[a\] is raised to the negative power \[ - b\].
Then, \[{a^{ - b}}\] is equal to the reciprocal of the number \[a\] raised to the positive power \[b\], that is \[{a^{ - b}} = {\left( {\dfrac{1}{a}} \right)^b}\].
Substituting \[a = \dfrac{4}{5}\] and \[b = 8\] in the equation, we get
\[ \Rightarrow {\left( {\dfrac{4}{5}} \right)^{ - 8}} = {\left( {\dfrac{1}{{\dfrac{4}{5}}}} \right)^8}\]
Simplifying the expression, we get
\[ \Rightarrow {\left( {\dfrac{4}{5}} \right)^{ - 8}} = {\left( {\dfrac{5}{4}} \right)^8}\]
\[\therefore \] We have expressed \[{\left[ {{{\left( {\dfrac{4}{5}} \right)}^{ - 2}}} \right]^4}\] as the rational number \[\dfrac{5}{4}\] raised to the positive exponent 8.
Now, it is given that the answer is \[{\left( {\dfrac{5}{4}} \right)^m}\].
This means that \[{\left( {\dfrac{5}{4}} \right)^8} = {\left( {\dfrac{5}{4}} \right)^m}\].
Comparing the terms on both sides, we get
\[8 = m\]
\[\therefore \]The value of \[m\] is 8.

Note: We compared the terms on both sides of the equation \[{\left( {\dfrac{5}{4}} \right)^8} = {\left( {\dfrac{5}{4}} \right)^m}\]. The comparison is possible because if two numbers with the same base are equal, then their exponents are also equal. This means that if \[{a^m} = {a^n}\], then \[m = n\].