Question

How do you show that \[{512^{\dfrac{2}{3}}} = {16^{\dfrac{3}{2}}}\] by using the properties of exponents?

Answer 1

Hint: We have to show that \[{512^{\dfrac{2}{3}}} = {16^{\dfrac{3}{2}}}\] by using the properties of exponents. For this, first we will take the LHS and we will rewrite it using a property of exponents i.e., \[{\left( {{r^m}} \right)^n} = {r^{m \times n}}\] and then we will simplify it. Then, we will take the RHS and again we will use the property \[{\left( {{r^m}} \right)^n} = {r^{m \times n}}\] and then we will simplify it. If the result of LHS and RHS are equal then \[{512^{\dfrac{2}{3}}} = {16^{\dfrac{3}{2}}}\].

Formula used:
To simplify a given expression, we need to follow some rules of exponents:
\[(1)\] When bases are the same and the operation between the numbers is multiplication, then the power of these numbers are just added i.e., \[{r^m} \times {r^n} = {r^{\left( {m + n} \right)}}\].
\[(2)\] When a number is raised by another number, the powers are multiplied with each other i.e., \[{\left( {{r^m}} \right)^n} = {r^{m \times n}}\].
\[(3)\] When two different numbers have the same base, each base will be raised to the same power i.e., \[{\left( {rm} \right)^s} = {r^s} \times {m^s}\].
\[(4)\] When the bases are different and the operation between the bases is division, then the powers of these bases are raised to each base individually i.e., \[{\left( {\dfrac{r}{m}} \right)^s} = \dfrac{{{r^s}}}{{{m^s}}}\].
\[(5)\] When the bases are the same and the operation between the bases is division, then the power of the denominator is subtracted from the power of the numerator i.e., \[\dfrac{{{r^n}}}{{{r^m}}} = {r^{n - m}}\].
\[(6)\] When the bases are the same and the operation between the bases is division, then the power of the numerator is subtracted from the power of the denominator i.e., \[\dfrac{{{r^n}}}{{{r^m}}} = \dfrac{1}{{{r^{m - n}}}}\].

Complete step by step solution:
We have to show that \[{512^{\dfrac{2}{3}}} = {16^{\dfrac{3}{2}}}\] by using the properties of exponents.
Here, LHS \[ = {512^{\dfrac{2}{3}}}\] and RHS \[ = {16^{\dfrac{3}{2}}}\].
Taking LHS first, we have
\[ \Rightarrow {\text{LHS}} = {512^{\dfrac{2}{3}}}\]
As we know, \[{\left( {{r^m}} \right)^n} = {r^{m \times n}}\]. Using this, we get
\[ \Rightarrow {\text{LHS}} = {\left( {{{512}^{\dfrac{1}{3}}}} \right)^2}\]
As \[\left( {{{512}^{\dfrac{1}{3}}}} \right) = 8\]. So, we get
\[ \Rightarrow {\text{LHS}} = {\left( 8 \right)^2}\]
\[ = 64\]
Now, taking RHS, we have
\[ \Rightarrow {\text{RHS}} = {16^{\dfrac{3}{2}}}\]
Again, using the property \[{\left( {{r^m}} \right)^n} = {r^{m \times n}}\], we get
\[ \Rightarrow {\text{RHS}} = {\left( {{{16}^{\dfrac{1}{2}}}} \right)^3}\]
As, \[{16^{\dfrac{1}{2}}} = 4\]. Using this, we get
\[ \Rightarrow {\text{RHS}} = {\left( 4 \right)^3}\]
\[ = 64\]
We can see clearly that \[{\text{LHS}} = {\text{RHS}} = 64\].
Therefore, \[{512^{\dfrac{2}{3}}} = {16^{\dfrac{3}{2}}}\].

Note:
Exponents are also called power. It is a number which indicates the number of times multiplication is to be performed. An exponent can be a positive or negative number. Any number, \[{a^b}\] is read as ‘\[a\] is raised to the power \[b\]’. Also note that when \[a\] is raised to the power \[0\], it is equal to \[1\] i.e., \[{a^0} = 1\].