Question

Express \[\dfrac{2}{3}\sqrt {1800} \] in its simplest form.

Answer 1

Hint: Here we need to simplify the given expression. This expression consists of roots and fractions. So we will first convert the root in the form of power, then we will use the properties of the exponents to simplify it further. After using the different properties of the exponents and using the different mathematical operations, we get the required answer.

Complete step-by-step answer:
The given expression is \[\dfrac{2}{3}\sqrt {1800} \]
We will write the number 1800 as a product of factors.
\[\dfrac{2}{3}\sqrt {1800} = \dfrac{2}{3}\sqrt {2 \times 3 \times 3 \times 10 \times 10} \]
We know that when the exponentials with the same base are multiplied then their power gets added.
Using the above mentioned property, we get
\[ \Rightarrow \dfrac{2}{3}\sqrt {1800} = \dfrac{2}{3}\sqrt {2 \times {3^2} \times {{10}^2}} \]
Now, we will write the root used in the expression in the form of power.
\[ \Rightarrow \dfrac{2}{3}\sqrt {1800} = \dfrac{2}{3}{\left( {2 \times {3^2} \times {{10}^2}} \right)^{\dfrac{1}{2}}}\]
We know from properties of the exponents that \[{\left( {a \cdot b} \right)^2} = {a^2} \cdot {b^2}\].
Using this property of exponents in the above equation, we get
\[ \Rightarrow \dfrac{2}{3}\sqrt {1800} = \dfrac{2}{3} \times {2^{\dfrac{1}{2}}} \times {3^{2 \times \dfrac{1}{2}}} \times {10^2}^{ \times \dfrac{1}{2}}\]
On further simplifying the power, we get
\[ \Rightarrow \dfrac{2}{3}\sqrt {1800} = \dfrac{2}{3} \times {2^{\dfrac{1}{2}}} \times 3 \times 10\]
On multiplying the terms, we get
\[ \Rightarrow \dfrac{2}{3}\sqrt {1800} = {2^{\dfrac{1}{2}}} \times 20\]
Now, we will convert the power into square roots. Therefore, we get
\[ \Rightarrow \dfrac{2}{3}\sqrt {1800} = \sqrt 2 \times 20\]
On further simplification, we get
\[ \Rightarrow \dfrac{2}{3}\sqrt {1800} = 20\sqrt 2 \]
Hence, the simplified value of the given expression i.e. \[\dfrac{2}{3}\sqrt {1800} \] is \[20\sqrt 2 \] .

Note: Here, exponents are used to represent the number of times the given number is multiplied with itself. When the exponentials with the same base are multiplied then their power gets added. Similarly if the exponential with the same base is divided by another exponential with the same then their power gets subtracted. This is also known as the addition and subtraction properties of the exponentials.