Question

Simplify ${(256)^{ - ({4^{ -\dfrac{ 3}{2}}})}}$

Answer 1

Hint: We first solve the power that exists in the power of the given term. Use the formula of exponents to cancel the terms within power so as to make the solution easier. Write prime factorization of the number in the base and solve the term using the formula of exponents.

Formula used:

* Formula of exponents states that powers can be multiplied such that ${({x^m})^{\dfrac{1}{n}}} = {x^{m \times \dfrac{1}{n}}} = {x^{\dfrac{m}{n}}}$

* Prime factorization of a number is writing the number in multiples of its factors where all factors are prime numbers.

* When the base is same powers can be added i.e. ${a^m} \times {a^n} = {a^{m + n}}$

* If the power has negative sign, then we take reciprocal of the term i.e. ${a^{ - 1}} = \dfrac{1}{a}$

Complete step by step answer:

We have to solve for the value of ${(256)^{ - ({4^{ - \dfrac{3}{2}}})}}$ ……….… (1)

Firstly, we solve the term in the power

The term in the power is ${4^{\dfrac{{ - 3}}{2}}}$

We can write $4 = 2 \times 2$

Since base is same we can add the powers

$ \Rightarrow 4 = {2^2}$

So, the term in power becomes

$ \Rightarrow {4^{\dfrac{{ - 3}}{2}}} = {({2^2})^{\dfrac{{ - 3}}{2}}}$

Use the formula of exponents ${({x^m})^{\dfrac{1}{n}}} = {x^{\dfrac{m}{n}}}$

$ \Rightarrow {4^{\dfrac{{ - 3}}{2}}} = {(2)^{2 \times \dfrac{{ - 3}}{2}}}$

Cancel same terms from numerator and denominator in the power

$ \Rightarrow {4^{\dfrac{{ - 3}}{2}}} = {(2)^{ - 3}}$

Since there is negative sign in the power, we take reciprocal of the term

$ \Rightarrow {4^{\dfrac{{ - 3}}{2}}} = \dfrac{1}{{{2^3}}}$

Open the expansion in denominator

$ \Rightarrow {4^{\dfrac{{ - 3}}{2}}} = \dfrac{1}{{2 \times 2 \times 2}}$

$ \Rightarrow {4^{\dfrac{{ - 3}}{2}}} = \dfrac{1}{8}$

Substitute the value of ${4^{\dfrac{{ - 3}}{2}}} = \dfrac{1}{8}$ in equation (1)

$ \Rightarrow {(256)^{ - ({4^{ - \dfrac{3}{2}}})}} = {(256)^{\dfrac{{ - 1}}{8}}}$ ……….… (2)

Now we write the prime factorization of 256

$256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

Since base is same, add the power

$ \Rightarrow 256 = {2^8}$

Substitute the value of $256 = {2^8}$ in equation (2)

$ \Rightarrow {(256)^{ - ({4^{ - \dfrac{3}{2}}})}} = {({2^8})^{\dfrac{{ - 1}}{8}}}$

Use the formula of exponents ${({x^m})^{\dfrac{1}{n}}} = {x^{\dfrac{m}{n}}}$

$ \Rightarrow {(256)^{ - ({4^{ - \dfrac{3}{2}}})}} = {(2)^{8 \times \dfrac{{ - 1}}{8}}}$

Cancel same terms from numerator and denominator in the power

$ \Rightarrow {(256)^{ - ({4^{ - \dfrac{3}{2}}})}} = {(2)^{ - 1}}$

Since there is negative sign in the power, we take reciprocal of the term

$ \Rightarrow {(256)^{ - ({4^{ - \dfrac{3}{2}}})}} = \dfrac{1}{2}$

$\therefore $ The value of ${(256)^{ - ({4^{ - \dfrac{3}{2}}})}}$ is $\dfrac{1}{2}$.

Note: Students might make mistakes solving the power that exists in the power as they find it difficult to break the term $ - \dfrac{3}{2}$. Keep in mind we should always write the term in the base in the smallest and simplest form after collecting its powers so we can cancel as many terms in the power as possible.