Question

How do you simplify ${(4{a^3})^2}$ and write it using only positive exponents?

Answer 1

Hint: In this question, we want to multiply the exponents of the expression. The power of a number is known as the exponent. For that, we will apply the power to power property which states that when a number is raised by another number, the powers are multiplied with each other. Therefore, the formula for the power to power property is: ${\left( {{x^a}} \right)^b} = {x^{a \times b}}$ . Here, x is the base and a and b are the exponents.

Complete step-by-step solution:
In this question, we want to simplify the given expression.
$ \Rightarrow {(4{a^3})^2}$
Here, we want to multiply the exponents of the above expression.
According to the power to power property, when a number is raised by another number, the powers are multiplied with each other.
Therefore, the formula for the power to power property is: ${\left( {{x^a}} \right)^b} = {x^{a \times b}}$.
In this question, we want to multiply the exponents a and b. Here, the base is the same i.e. x and exponents are 3 and 2. So, we will multiply the exponents 3 and 2. The answer will be 6.
Substitute the values in the power to the power property formula.
$ \Rightarrow {(4{a^3})^2} = 4{a^{3 \times 2}}$
The multiplication of 2 and 3 is 6.
 $ \Rightarrow {(4{a^3})^2} = 4{a^6}$
Hence, the solution of ${(4{a^3})^2}$is $4{a^6}$.

Note: Some exponent properties are as below.
If the bases are x and y, and the exponents are a and b.
a)Product of power property: ${x^a} \times {x^b} = {x^{a + b}}$
b). Power to a power property: ${\left( {{x^a}} \right)^b} = {x^{ab}}$
c). Power of a product property: ${\left( {xy} \right)^a} = {x^a}{y^a}$
d). The quotient of power property: $\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}$
e). Power of a quotient property: ${\left( {\dfrac{x}{y}} \right)^a} = \dfrac{{{x^a}}}{{{y^a}}}$
Negative exponents are the reciprocals of the positive exponents.
Therefore, ${x^{ - a}} = \dfrac{1}{{{x^a}}}$ and
 ${x^a} = \dfrac{1}{{{x^{ - a}}}}$