Question

How do you write $4{{a}^{-6}}$ using only positive exponents?

Answer 1

Hint: We know that the exponent of a number is the number to which the base raised. So, we can say that the exponent is said to be positive if the sign of the exponent is positive. We should remember the identity ${{x}^{-n}}=\dfrac{1}{{{x}^{n}}}.$

Complete step by step answer:
Consider a number of the form ${{a}^{b}}.$ The digit $a$ is called the base of the number and the digit $b$ is called the exponent of the number.
So, we can say that the exponent is a number to which the base is raised. So, when the power of a number is positive, we say the number has positive exponents.
Let us consider the given problem.
We are asked to write $4{{a}^{-6}}$ using only positive exponents.
We need to change the negative exponent to positive. For that, we can use the identity ${{x}^{-n}}=\dfrac{1}{{{x}^{n}}}.$
Consider the term containing negative exponent. That is, ${{a}^{-6}}.$
By using the above identity, we can convert this term to a term with positive exponent. According to the above identity, we just have to bring this term from the numerator to the denominator.
So, we will get ${{a}^{-6}}=\dfrac{1}{{{a}^{6}}}.$
In the next step, we are going to substitute this in our problem. From this, we will yield $4{{a}^{-6}}=4\dfrac{1}{{{a}^{6}}}.$

Hence the equivalent term $4{{a}^{-6}}$ using only the positive exponents is $\dfrac{4}{{{a}^{6}}}.$

Note: Suppose that we are multiplying and dividing the given quantity $4{{a}^{-6}}$ with ${{a}^{6}}.$ So, we will get $\dfrac{4{{a}^{-6}}{{a}^{6}}}{{{a}^{6}}}.$ We will use the identity ${{x}^{-n}}{{x}^{m}}={{x}^{-n+m}}.$ We will get $\dfrac{4{{a}^{-6}}{{a}^{6}}}{{{a}^{6}}}=\dfrac{4{{a}^{-6+6}}}{{{a}^{6}}}=\dfrac{4{{a}^{0}}}{{{a}^{6}}}=\dfrac{4\times 1}{{{a}^{6}}}=\dfrac{4}{{{a}^{6}}}.$