Question

How do you simplify ${x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}}$?

Answer 1

Hint:In this question we need to simplify expression ${x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}}$. To simplify questions of these types we use laws of exponents or powers. Exponents are very useful in writing very small or big numbers very efficiently. Here we will use the law of exponent in which if two numbers with the same base are in product then their exponents will add up .

Complete step by step solution:
Let us try to solve question in which we are asked to simplify given expression ${x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}}$. To solve this we have to use the law of exponents. Before solving this we are required to have known about the exponents and its properties.
A number is in exponent if it is written as ${a^b}$ where $a$ is called base $b$ is power or exponent of the number. To solve this we will use this property of exponents${a^b} \cdot {a^c} = {a^{b + c}}$. Let’s come back to our problem of simplifying ${x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} \cdot
{x^{\dfrac{1}{3}}}$.
To simplify this we will use the associative property of multiplication.
We can write this as,
${x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} = ({x^{\dfrac{1}{3}}}
\cdot {x^{\dfrac{1}{3}}}) \cdot {x^{\dfrac{1}{3}}}$
Now, applying the law of exponents ${a^b} \cdot {a^c} = {a^{b + c}}$ to the above equation. We get,
$
{x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} = ({x^{\dfrac{1}{3}}} \cdot
{x^{\dfrac{1}{3}}}) \cdot {x^{\dfrac{1}{3}}} \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {x^{\dfrac{1}{3} + \dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} \\
$
Now, performing fraction addition in above equation, we get
\[{x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}}\, =
\,{x^{\dfrac{2}{3}}} \cdot {x^{\dfrac{1}{3}}}\] $eq(1)$
Now, again applying ${a^b} \cdot {a^c} = {a^{b + c}}$ to the equation, we get
\[{x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}}\, = \,{x^{\dfrac{2}{3}
+ \dfrac{1}{3}}}\]
Now, performing fraction addition in above equation, we get
\[{x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}}\, = \,{x^1} = x\]
Hence the expression ${x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}} \cdot {x^{\dfrac{1}{3}}}$simplifies to $x$.

Note: Solving these types of questions requires the knowledge of laws of exponents. Generally Students make mistakes in these questions while performing calculations, so be careful while doing calculations. Exponents are generally used to write very big or small numbers like the distance between the earth and the sun, the size of an atom.