Question

Find the multiplicative inverse of \[{\left( { - 7} \right)^{ - 2}} \div {\left( {90} \right)^{ - 1}}\]

Answer 1

Hint: Here we are asked to find the multiplicative inverse of the given expression with exponent terms. First, we will try to simplify the given expression then we will find its multiplicative inverse. The multiplicative inverse of a term is nothing but the reciprocal of it. That is, a multiplicative inverse of a number is the reciprocal of that number.
Formula: Some formulas that we need to know:
\[{a^{ - x}} = \dfrac{1}{{{a^x}}}\]
If \[\dfrac{p}{q}\] and \[\dfrac{r}{s}\] are any rational numbers then \[\dfrac{p}{q} \div \dfrac{r}{s} = \dfrac{p}{q} \times \dfrac{s}{r}\]

Complete step by step answer:
It is given that \[{\left( { - 7} \right)^{ - 2}} \div {\left( {90} \right)^{ - 1}}\] we aim to find the multiplicative inverse of the given expression.
Before finding the multiplicative inverse, we will simplify the given expression.
Consider the given expression, \[{\left( { - 7} \right)^{ - 2}} \div {\left( {90} \right)^{ - 1}}\] we can see that the terms in this expression are exponents terms. Since the exponent terms have negative power, we will convert it to positive power by using the exponent formula.
Using the formula \[{a^{ - x}} = \dfrac{1}{{{a^x}}}\] the given expression can be written as
\[ \Rightarrow \dfrac{1}{{{{\left( { - 7} \right)}^2}}} \div \dfrac{1}{{{{\left( {90} \right)}^1}}}\]
On simplifying it further we get
\[ \Rightarrow \dfrac{1}{{{7^2}}} \div \dfrac{1}{{90}}\]
We know that if \[\dfrac{p}{q}\] and \[\dfrac{r}{s}\] are any rational numbers then \[\dfrac{p}{q} \div \dfrac{r}{s} = \dfrac{p}{q} \times \dfrac{s}{r}\]. That is, we cannot divide directly on rational numbers so we will convert them to multiplication to solve them.
\[ \Rightarrow \dfrac{1}{{{7^2}}} \times 90\]
\[ \Rightarrow \dfrac{{90}}{{{7^2}}}\]
On simplifying it further we get
\[ \Rightarrow \dfrac{{90}}{{49}}\]
\[{\left( { - 7} \right)^{ - 2}} \div {\left( {90} \right)^{ - 1}} = \dfrac{{90}}{{49}}\]
Now we have Simplified the given expression to a simple rational number. But we aim to find the multiplicative inverse of it. We know that multiplicative inverse is nothing but reciprocal.
The reciprocal of the number \[\dfrac{{90}}{{49}}\] is \[\dfrac{{49}}{{90}}\].
Therefore, the multiplicative inverse of the expression \[{\left( { - 7} \right)^{ - 2}} \div {\left( {90} \right)^{ - 1}}\] is \[\dfrac{{49}}{{90}}\].

Note:
The exponent is nothing but the no. of times a number multiplied by itself i.e., \[a \times a \times a = {a^3}\] here \[a - \]base and \[3 - \]exponent. Also \[{a^3}\] is called exponential form or power notation. We can also say it as \[a\] raised to the power three.