Question

How do you simplify $\dfrac{{{5^4} \times {5^7}}}{{{5^8}}}$?

Answer 1

Hint: We use the concept of exponents and solve the value using the property which states that the powers can be multiplied if we have an element like this i.e. \[{({x^m})^n} = {x^{mn}}\]. Multiply the terms in the power and use the concept of reciprocal to write the final answer.
* If the power has negative sign, then we take reciprocal of the term i.e. \[{a^{ - 1}} = \dfrac{1}{a}\]

Complete step by step answer:
We have to simplify the value \[{\left( {3{x^4}} \right)^{ - 2}}\]
Here we use the identity of exponent i.e. \[{({x^m})^n} = {x^{mn}}\]
Multiply the terms in the power outside the bracket to each power of term inside the bracket
\[ \Rightarrow {\left( {3{x^4}} \right)^{ - 2}} = {3^{ - 2}}{x^{4 \times ( - 2)}}\]
Calculate the product of terms in the power
\[ \Rightarrow {\left( {3{x^4}} \right)^{ - 2}} = {3^{ - 2}}{x^{ - 8}}\]
Now we have a negative power on the right hand side of the equation i.e. -8.
We use the method of reciprocal to write the power in the right hand side of the equation as positive power by writing the term as reciprocal.
\[ \Rightarrow {\left( {3{x^4}} \right)^{ - 2}} = \dfrac{1}{{{3^2}{x^8}}}\]
Calculate square of 3 i.e. multiply 3 with itself i.e. \[3 \times 3 = 9\]
\[ \Rightarrow {\left( {3{x^4}} \right)^{ - 2}} = \dfrac{1}{{9{x^8}}}\]

The value of \[{\left( {3{x^4}} \right)^{ - 2}}\] on simplification is \[\dfrac{1}{{9{x^8}}}\]

Note: Many students make the mistake of not taking the power with constant terms when they open the bracket i.e. when they multiply the power from outside the bracket with power inside the bracket; they forget to take the same power to 3. Keep in mind we put on the same power to all terms inside the bracket when we open the bracket. Also, when taking reciprocal, students again leave constant term in the numerator, since the square of constant term also has negative power we will take its reciprocal as well and shift it to the denominator.