Question

How do you simplify the expression ${{\left( {{x}^{2}} \right)}^{4}}$

Answer 1

Now to simplify the given expression we will use the power law of indices. According to multiplication law of indices we have ${{\left( {{x}^{m}} \right)}^{n}}$ as ${{x}^{m\times n}}$ hence using this in the given expression we will have a simplified expression.

Complete step-by-step solution:
Now to solve the given equation we will first understand the concept of indices.
Now Indices are nothing but a number raised to another number.
The number to which another number is raised is called base and the number which is raised is called the power of base.
For example consider ${{2}^{3}}$ here 2 is the base and 3 is the power of base.
Now let us understand what does ${{2}^{3}}$ means. Basically it means multiplication of 2, 3 times.
Hence we expand the indice as ${{2}^{3}}=2\times 2\times 2$ .
Now that we know what indices are, we can understand some basic laws of indices.
When we have 0 in power the value of the expression is 1.
Hence ${{x}^{0}}=1$
Now similarly we have the following laws in indices.
Multiplication law: ${{a}^{m}}{{a}^{n}}={{a}^{m+n}}$
Division law: $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
Power law: ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$
Now consider the given expression ${{\left( {{x}^{2}} \right)}^{4}}$
Now we know by power law of indices that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$
Hence using this we can say that ${{\left( {{x}^{2}} \right)}^{4}}={{x}^{2\times 4}}={{x}^{8}}$
Hence, we have ${{\left( {{x}^{2}} \right)}^{4}}={{x}^{8}}$
Hence the given equation is simplified.

Note: Now note that we can also solve the question by just using definition. Now we know that ${{x}^{2}}=\left( x\times x \right)$ hence substituting this we get ${{\left( {{x}^{2}} \right)}^{4}}={{\left( x\times x \right)}^{4}}$ Now again using the definition we get, ${{\left( x\times x \right)}^{4}}=\left( x\times x \right)\times \left( x\times x \right)\times \left( x\times x \right)\times \left( x\times x \right)$ and hence ${{\left( x\times x \right)}^{4}}={{x}^{8}}$ . Similarly we can show other properties by expanding the equation. Also note that power can be positive, negative or fraction.