Question

Find quotient: \[( - 15{x^8}) \div (5{x^5})\].

Answer 1

Hint: In exponents, powers of the same variable get added or subtracted. Exponents and powers are used to represent very large numbers or very small numbers in a simplified manner. For example, if we have to show 3\[ \times \]3\[ \times \]3\[ \times \]3 in a simplified manner, then we can write it as 34, where 4 is the exponent and $3$ is the base.

Formula used:
 \[\dfrac{{{{(x)}^a}}}{{{{(x)}^b}}} = {(x)^{a - b}}\]
\[{(x)^a} \times {(x)^b} = {(x)^{a + b}}\]

Complete step by step answer:
(1) Given: \[( - 15{x^8}) \div 5{(x)^5}\] or \[\dfrac{{ - 15{x^8}}}{{5({x^5})}}\]
(2) Take \[{\left( x \right)^5}\]to the numerator and simplify the question, we get
$ = - 15 \div 5\,{x^8} \times {x^{ - 5}}$
\[ = - 3.{(x)^8}.{(x)^{ - 5}}\]
(2) We know in exponents under multiplication, powers get added when bases are the same.
\[ = - 3{(x)^{8 + ( - 5)}}\]
\[ = - 3{(x)^{8 - 5}}\]
\[ = - 3{(x)^3}\]
\[ = - 3{x^3}\]
This is the required quotient of the given division.

Additional information: Basically, a power is an expression that shows repeated multiplication of the same number or factor. The value of the exponent is based on the number of times the base is multiplied to itself.

Note: Students must take care while using the signs in exponent as when we have two different signs between two numbers then in general we perform subtraction. Also, when we write the denominator in a numerator then a negative sign is added to its exponent or power.