Question

How do you simplify $3{\sqrt x ^{15}}$ ?

Answer 1

Hint: In this question, we are given that $\sqrt x $ is raised to the power $15$ . When a number is multiplied with itself 2 times, the number obtained is known as the square of the given number. So square root is defined as the number whose square we found. For example, 2 multiplied with itself gives 4, so 4 is the square of 2 and 2 is known as the square root of 4. $\sqrt x $ can also be written as ${x^{\dfrac{1}{2}}}$ . We will use the law of exponent which states that when a number raised to some power is again raised to another power then keeping the base same, we multiply the powers, that is, ${({a^x})^y} = {a^{x \times y}}$ . This way we can find the simplified value of the given expression.

Complete step-by-step solution:
We have to simplify $3{\sqrt x ^{15}}$
We can write it as –
$3{\sqrt x ^{15}} = 3{[{(x)^{\dfrac{1}{2}}}]^{15}}$
We know that ${({a^x})^y} = {a^{x \times y}}$
So, we get –
$
  3{\sqrt x ^{15}} = 3{x^{\dfrac{1}{2} \times 15}} \\
   \Rightarrow 3{\sqrt x ^{15}} = 3{x^{\dfrac{{15}}{2}}} \\
 $
Hence, the simplified form of $3{\sqrt x ^{15}}$ is $3{x^{\dfrac{{15}}{2}}}$ .


Note: We are given an exponential function $3{\sqrt x ^{15}}$ in this question. $\sqrt x $ is raised to the power 15, so $\sqrt x $ is called the base and $15$ is called its power. When a number is raised to the power “n”, it means that the number is multiplied with itself “n” times, for example, let ${a^n}$ be an exponential function, it means that the number “a” is multiplied with itself “n” times. So, $3{\sqrt x ^{15}}$ means that $\sqrt x $ is multiplied with itself 15 times and the expression obtained is multiplied by 3. This way we can find the value of any number that is raised to the power of some other number.