Question

If ${\left( {\dfrac{5}{3}} \right)^{ - 15}} \times {\left( {\dfrac{5}{3}} \right)^{11}} = {\left( {\dfrac{5}{3}} \right)^{8x}}$, then $x = ?$

Answer 1

Hint : Simplification is the process of writing the given algebraic expression effectively and most comfortably to understand without affecting the original expression. Moreover, various steps are involved to simplify an expression. Here, we need to apply the formula and some simplifications to the given equation.

Formula used:
${a^m}{a^n} = {a^{m + n}}$

Complete step-by-step solution:
Here, the given equation is ${\left( {\dfrac{5}{3}} \right)^{ - 15}} \times {\left( {\dfrac{5}{3}} \right)^{11}} = {\left( {\dfrac{5}{3}} \right)^{8x}}$
We need to calculate the value of $x$
${\left( {\dfrac{5}{3}} \right)^{ - 15}} \times {\left( {\dfrac{5}{3}} \right)^{11}} = {\left( {\dfrac{5}{3}} \right)^{8x}}$
Since the bases of the equation on the left-hide side are equal, we shall use the formula ${a^m}{a^n} = {a^{m + n}}$ on the above equation.
$ \Rightarrow {\left( {\dfrac{5}{3}} \right)^{ - 15 + 11}} = {\left( {\dfrac{5}{3}} \right)^{8x}}$
$ \Rightarrow {\left( {\dfrac{5}{3}} \right)^{ - 4}} = {\left( {\dfrac{5}{3}} \right)^{8x}}$
Since the base of the above equation is equal on both sides, we can compare the exponents/power of the base.
$ \Rightarrow - 4 = 8x$
$ \Rightarrow x = - \dfrac{4}{8}$
$ \Rightarrow x = - \dfrac{1}{2}$
Therefore, the required value of $x$ for the given equation is $ - \dfrac{1}{2}$ .
Additional information:
Simplification of an expression is the process of changing the expression effectively without changing the meaning of an expression.
 In some questions, we need to apply the BODMAS rule (i.e.) we need to calculate the brackets first and then orders, then division or multiplication, and finally we need to add or subtract.
Moreover, various steps are involved to simplify an expression. Some of the steps are listed below:
If the given expression contains like terms, we need to combine them.
Example: $3x + 2x + 4 = 5x + 4$
We need to split an expression into factors (i.e) the process of finding the factors for the given expression.
Example: ${x^2} + 4x + 3 = (x + 3)(x + 1)$
We need to expand an algebraic expression (i.e) we have to remove the respective brackets of an expression.
Example: $3(a + b) = 3a + 3b$.
We need to cancel out the common terms in an expression.

Note:Since it is difficult to expand the powers, we just apply the formula of exponent in this case. To apply the formula of exponent, we must check whether the bases of expression where we like to apply the formula are equal or not. If the bases are not equal, we cannot apply the formula, and also we cannot compare the respective powers.