Question

How do you simplify \[\dfrac{-6\times {{10}^{-5}}}{-2\times {{10}^{3}}}\]?

Answer 1

Hint: In order to simplify the given expression i.e. \[\dfrac{-6\times {{10}^{-5}}}{-2\times {{10}^{3}}}\], first we need to write the terms in the expression in the form of factors. After that applying the law of exponent which states that the negative sign of the base can be represented as reciprocal with the positive exponent i.e. \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\]. Then simplifying the given expression and again using the other law of exponent i.e. \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]. Simplifying the expression, we will get the required answer of the given question.

Formula used:
The negative sign of the base can be represented as reciprocal with the positive exponent i.e. \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\]
From the law of product with the same base that while multiplying the exponential terms with the same base, then we will add the exponent’s i.e. \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\].

Complete step by step answer:
We have given that,
\[\Rightarrow \dfrac{-6\times {{10}^{-5}}}{-2\times {{10}^{3}}}\]
The above expression can be written as,
\[\Rightarrow \dfrac{\left( -2\times 3 \right)\times {{10}^{-5}}}{-2\times {{10}^{3}}}\]
As we know that,
The negative sign of the base can be represented as reciprocal with the positive exponent i.e. \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\]
Therefore,
In the above expression, \[{{10}^{-5}}=\dfrac{1}{{{10}^{5}}}\]
Now,
We can write the above expression as,
\[\Rightarrow \dfrac{-2\times 3}{-2\times {{10}^{3}}\times {{10}^{5}}}\]
Cancelling out the common terms, we will get
\[\Rightarrow \dfrac{3}{{{10}^{3}}\times {{10}^{5}}}\]
As we know that,
From the law of product with the same base that while multiplying the exponential terms with the same base, then we will add the exponent’s i.e. \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\].
Therefore,
Applying this law of exponent in the denominator, we will get
\[\Rightarrow \dfrac{3}{{{10}^{3+5}}}\]
Simplifying the exponent in the denominator, we will get
\[\Rightarrow \dfrac{3}{{{10}^{8}}}\]
As we know that,
We can write \[\dfrac{1}{{{10}^{8}}}={{10}^{-8}}\],
Therefore,
\[\Rightarrow \dfrac{3}{{{10}^{8}}}=3\times {{10}^{-8}}\]

Therefore,
The value of \[\dfrac{-6\times {{10}^{-5}}}{-2\times {{10}^{3}}}\] is equal to \[3\times {{10}^{-8}}\].
Hence, this is the required solution.

Note: While solving exponential problems we need to be very careful while doing the calculation part to avoid making any type of calculation mistake. In order to solve these types of questions, the conceptual knowledge about laws of exponents and powers is required. Students should always keep in mind the various laws of exponents and powers in order to solve these types of questions.