Question

The value of ${{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}$ is
$\begin{align}
  & a)64 \\
 & b)256.25 \\
 & c)16 \\
 & d)4 \\
\end{align}$

Answer 1

Hint: Now to solve this we will first use the property ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ Now to the obtained expression we will convert the power from decimals to fraction and reduce it in its lowest form. Now we know that ${{x}^{\dfrac{p}{q}}}=\sqrt[q]{{{x}^{p}}}$ hence we will simplify the expression and find the value of ${{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}$ .

Complete step by step answer:
Now let us understand the meaning of ${{x}^{n}}$ . ${{x}^{n}}=x\times x\times ......n\text{ times}$ . Hence ${{x}^{n}}$ is multiplication of the number x, n times. Now let us understand what happens when the power is a fraction. Consider the number ${{x}^{\dfrac{p}{q}}}$ then we have ${{x}^{\dfrac{p}{q}}}=\sqrt[q]{{{x}^{p}}}$ .
Now to solve equations in indices we have some properties of indices.
${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
$\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$
Hence we use the appropriate identity to solve the equations.
Now consider the given expression ${{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}$ .
Now using the property ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ we get,
$\begin{align}
  & {{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}={{\left( 256 \right)}^{0.16+0.09}} \\
 & \Rightarrow {{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}={{\left( 256 \right)}^{0.25}} \\
\end{align}$
Now we know that $0.25=\dfrac{25}{100}$ and $\dfrac{25}{100}=\dfrac{1}{4}$ .
Hence substituting these values we get,
${{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}={{\left( 256 \right)}^{\dfrac{1}{4}}}$
Now we know that ${{x}^{\dfrac{p}{q}}}=\sqrt[q]{{{x}^{p}}}$ hence we get, ${{\left( 256 \right)}^{\dfrac{1}{4}}}=\sqrt[4]{256}$ .
Now let us factorize 256.
256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
256 = 4 × 4 × 4 × 4
Hence we can say that $\sqrt[4]{256}=4$ .
Hence ${{\left( 256 \right)}^{0.16}}\times {{\left( 256 \right)}^{0.09}}=4$

So, the correct answer is “Option d”.

Note: We have that for any real number x, ${{x}^{0}}=1$ .Now the power of a number can also be negative. For example let us say we have a term ${{x}^{-n}}$ . Now in this case ${{x}^{-n}}$ is written as $\dfrac{1}{{{x}^{n}}}$ . Now note that the even powers of indices are always positive. For example we have $-{{5}^{2}}=25$ and ${{5}^{2}}=25$ but for odd powers if the base is negative then the value is also negative. For example ${{\left( -2 \right)}^{3}}=-8$ .