Question

How do you fill in the blanks for the expression ${{r}^{?}}.{{r}^{12}}={{r}^{20}}?$

Answer 1

Hint: To answer this question, we need to know the basic concepts of exponents. To solve this question, we assume the blank to be filled to have a value x and apply the multiplication law of exponents to simplify for the left-hand side. We then equate this to the right-hand side and solve for x.

Complete step by step solution:
The given contains the expression ${{r}^{?}}.{{r}^{12}}={{r}^{20}}.$ Let us assume the value to be filled in the blank is x.
$\Rightarrow {{r}^{x}}.{{r}^{12}}={{r}^{20}}$
We now use the law of multiplication of exponents. This rule states that if we multiply two exponents with the same base but different powers, we get a result with the same base but the powers being added. This is expressed as ${{a}^{m}}\times {{a}^{n}}={{a}^{\left( m+n \right)}}.$
For the given question, we have the same base r. The powers on the left-hand side are x and 12. The product of these two exponents gives us an exponent whose power is equal to the sum of these two powers. Applying this law for the left-hand side,
$\Rightarrow {{r}^{x}}.{{r}^{12}}={{r}^{\left( x+12 \right)}}$
We now know that this expression on the left-hand side is equal to the expression on the right-hand side.
$\Rightarrow {{r}^{\left( x+12 \right)}}={{r}^{20}}$
We know that since the two sides of the equation are equal and they have the same bases, they must have the same powers too.
$\Rightarrow \left( x+12 \right)=20$
Subtracting both sides by 12,
$\Rightarrow \left( x+12 \right)-12=20-12$
The 12 on the left-hand side gets cancelled with -12 and we subtract 20 and 12 and write the result on the right-hand side.
$\Rightarrow x=8$

Hence, the number to filled in the blank is 8 and the final expression after filling in the blank is given as ${{r}^{8}}.{{r}^{12}}={{r}^{20}}.$

Note: Students need to have a good understanding in the topics of exponents and powers to solve this question. We can also solve this question by taking the ${{r}^{12}}$ term to the denominator on the right-hand side and applying the law of division of exponents $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{\left( m-n \right)}}.$