Question

How do you simplify ${(2{x^3})^4}$ ?

Answer 1

We will start by using the following exponent rule:
${({x^a})^b} = {x^{a \times b}}$. Mention all the terms. Then reduce the terms until they cannot be reduced any further. We will also use the rule ${x^a} = \dfrac{1}{{{x^{ - a}}}}$ to simplify the terms further.

Complete step by step answer:
We will start off by applying the exponent rule given by, ${({x^a})^b} = {x^{a \times b}}$.

$$= {(2{x^3})^4} \\\ = {2^4}{x^{3 \times 4}} \\\$$

Now we will simplify the terms which are within the parenthesis.

$$= {2^4}{x^{3 \times 4}} \\\ = 16{x^{12}} \\\$$

Hence, the value of the expression ${(2{x^3})^4}$ is $16{x^{12}}$.

Additional Information: The power rule tells us that to raise a power to a power, just multiply the exponents. The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. If any negative number is raised to a negative power equals its reciprocal raised to the opposite power. Any number raised to a power one equals itself. In the exponent product rule, when multiplying two powers that have the same base, you can add the exponents. A negative exponent means divides, because the opposite of multiplying is dividing. A fractional exponent means to take the ${n^{th}}$root. A quotient rule tells us that we can divide two powers with the same base by subtracting the exponents.

Note: While applying the rule, ${({x^a})^b} = {x^{a \times b}}$ make sure that you have considered terms properly so that the terms should get easier to solve. Also, while applying the rule, ${x^a} = \dfrac{1}{{{x^{ - a}}}}$ pay more attention to the sign of the powers. If the sign is missed whole the problem can be incorrect. Also, remember that the negative exponent means to divide, as the opposite of multiplying is dividing.