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How to simplify \[{\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}}\] in scientific notation?

seo-qna
Last updated date: 13th Jun 2024
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Answer
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Hint:
Here, we will use suitable exponent rules to simplify the given exponential expression. Then we will use the definition of scientific notation to rewrite the simplified value in the form of scientific notation. Scientific notation is a way of expressing a number that is too large or too small in a simpler form or in the decimal form.

Formula Used:
We will use the following formulas:
1) Power of a Product rule: \[{\left( {ab} \right)^m} = {a^m} \times {b^m}\]
2) Power rule for Exponents: \[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
3) Negative Exponent Rule: \[{\left( a \right)^{ - n}} = {\left( {\dfrac{1}{a}} \right)^n}\]
4) Product rule of exponents: \[{a^m} \times {a^n} = {a^{m + n}}\]
5) Zero Exponent: \[{a^0} = 1\]

Complete step by step solution:
We are given an exponential expression\[{\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}}\].
Now, by using the Power of Product rule \[{\left( {ab} \right)^m} = {a^m} \times {b^m}\], we get
\[ \Rightarrow {\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}} = {7.87^{ - 6}} \times {\left( {{{10}^{ - 1}}} \right)^{ - 6}}\]
Using the Power rule for exponents \[{\left( {{a^m}} \right)^n} = {a^{mn}}\], we get
\[ \Rightarrow {\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}} = {7.87^{ - 6}} \times {10^{ - 1 \times - 6}}\]
We know that the product of two negative integers is a positive integer. So, we get
\[ \Rightarrow {\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}} = {7.87^{ - 6}} \times {10^6}\]
Now, by using the Negative Exponent rule \[{\left( a \right)^{ - n}} = {\left( {\dfrac{1}{a}} \right)^n}\], we get
\[ \Rightarrow {\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}} = \left( {\dfrac{1}{{{{7.87}^6}}}} \right) \times {10^6}\]
Applying the exponents, we get
\[ \Rightarrow {\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}} = \left( {\dfrac{1}{{237601.0711}}} \right) \times {10^6}\]
Simplifying the expression, we get
\[ \Rightarrow {\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}} = 4.2087 \times {10^{ - 6}} \times {10^6}\]
Using the Product rule of exponents \[{a^m} \times {a^n} = {a^{m + n}}\], we get
\[ \Rightarrow {\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}} = 4.2087 \times {10^{ - 6 + 6}}\]
\[ \Rightarrow {\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}} = 4.2087 \times {10^0}\]
Now, by using the Zero exponent rule \[{a^0} = 1\], we get
\[ \Rightarrow {\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}} = 4.2087 \times 1\]
\[ \Rightarrow {\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}} = 4.2087\]
A number can be expressed in Scientific notation which is in the form \[N = a \times {10^n}\] where \[1 \le a < 10\] and \[n\] is an integer
The above equation is in the form of scientific notation where \[a\] is greater than \[1\] and less than \[10\].

Therefore, the value of the exponential expression \[{\left( {7.87 \times {{10}^{ - 1}}} \right)^{ - 6}}\] is \[4.2087\]

Note:
We know that an exponential expression is defined as an expression with the base and exponents. The number of digits that is moved from the decimal point(either left or right) is given by the power of \[10\] that is equal to \[{10^n}\]. If the decimal point is moved to the left, then the exponent \[n\] is positive and if the decimal point is moved to the right, then the exponent \[n\] is negative.