Question

How do you simplify $\dfrac{{{6}^{-6}}}{{{6}^{-5}}}$?

Answer 1

Hint: In order to solve any number having an exponent we first simplify the exponent then apply the exponent on the term to get the required answer. Here, we will simplify the given expression using the exponent rule. Then we will first use the negative exponent rule and then divide the terms to get the required answer.

Formula Used:
Negative Exponent Rule: \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\].

Complete step by step solution:
We are given an expression\[\dfrac{{{6^{ - 6}}}}{{{6^{ - 5}}}}\].
Let \[x\] be the given expression.
\[x = \dfrac{{{6^{ - 6}}}}{{{6^{ - 5}}}}\]
Negative Exponent Rule: \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\]
Now, by using the negative exponent rule, we get
$\Rightarrow x=\dfrac{\dfrac{1}{{{6}^{6}}}}{\dfrac{1}{{{6}^{5}}}}$
Now, by rewriting the equation, we get
$\Rightarrow x=\dfrac{{{6}^{5}}}{{{6}^{6}}}$
Now, by canceling the terms, we get
$\Rightarrow x=\dfrac{1}{6}$

Therefore, the value of the simplified expression \[\dfrac{{{6^{ - 6}}}}{{{6^{ - 5}}}}\]is \[\dfrac{1}{6}\].

Additional Information:
We know that the BODMAS rule states that the first operation has to be done which is in the brackets, next the operation applies on the indices or order, then it moves on to the division and multiplication, and then using addition and subtraction we will simplify the expression. If addition or subtraction and division or multiplication are in the same calculations, then it has to be done from left to right. An arithmetic expression is defined as an expression with the numbers and the arithmetic operators like plus, minus, etc.

Note:
We will follow the rules for exponents in order to simplify the exponential expression.
First, we will use the zero – exponent rule which says that if any number is raised to the power zero, then it is one i.e., \[{a^0} = 1\]
We will use the power rule next, which says that if any number is raised to the power and again raised to the power, then the exponents should be multiplied.
We will use the negative exponent rule, which says that the negative exponent in the numerator gets changed to the denominator, then the exponent becomes positive.