Question

Demonstrate that the product of two perfect squares is a perfect square as well.

Answer 1

Hint: we will first try to understand and use the law of exponents. In this question we will use the product to a power formula. The formula is \[{\left( {xy} \right)^a} = {x^a}{y^a}\] . We will take many examples to prove our theorem right.

Complete answer:
The law of exponents states that the bases are multiplied with a common whole power when two numbers of equal powers and distinct bases are multiplied.
${a^2} \times {b^2} = {\left( {ab} \right)^2}$
We can prove that the above theorem stands true by taking some examples
Let’s take a=3 and b=4
$ = {a^2} \times {b^2}$
$ = {3^2} \times {4^2}$
$ = 9 \times 16$
$ \Rightarrow 144 = {12^2}$
We can use any number and verify the theorem.
Hence, the product of two perfect squares is a perfect square.

Additional information:
Below we have discussed some type of law of exponents:
Product with the same base: When multiplying similar bases, that is, bases with the same exponents, the base remains the same and the exponents are added. Formula is \[{x^a} \times {x^b} = {x^{a + b}}\]
Power to power: Multiplication of powers occurs when a base with a power is raised to another power while the base stays the same. Formula is \[{\left( {{x^a}} \right)^b} = {x^{ab}}\]\[{x^a} \times {x^b} = {x^{a + b}}\]
Zero power: When you elevate anything to zero, it always equals one. for example: \[{x^0} = 1\]

Note: It is necessary to have a conceptual understanding of exponents and exponent laws. In order to tackle these sorts of issues, students need to always remember the numerous rules of exponents. Students might make mistakes while applying the law of exponent.