Question

How do you apply the “product of powers” property to simplify expressions?

Answer 1

Hint: While multiplying two powers with the same base, the exponents can be added, according to the exponent "product law." You have seen how it functions in the example. It's just a shortcut to incorporate the exponents! To lift a power to a power, simply multiply the exponents, according to the "power law."

Complete step by step solution:
The "product of powers" property says that when we multiply two powers which have the same base value, then we add the exponents of it.
The formula is ${a^m} \cdot {a^n} = {a^{m + n}}$
To understand the product of powers property more clearly, we shall look into a few examples,
${5^2} \cdot {5^4} = {5^{2 + 4}} = {5^6}$
${j^7} \cdot {j^3} = {j^{7 + 3}} = {j^{10}}$
${2^3} \cdot {2^5} \cdot {2^6} = {2^{3 + 5 + 6}} = {2^{14}}$
This property makes solving the expressions faster and easier. If we have to solve each term and then find the final answer, it will take a lot of time. So, this property is always preferred.

Additional information:
The Power of a Power Property states that you can find the power of a power by multiplying the exponents. That is, ${\left( {{a^m}} \right)^n} = {a^{mn}}$ for a non-zero real number a and two integers $m$ and $n$ . The Product of Powers Property says that when multiplying powers with the same bases, the exponents must be added.

Note:
If the powers of the same base are multiplied, then the exponents are added. This is the property of “product of powers.”
When powers of the same base are divided, the exponents are subtracted. This is the “quotient property.”
The “logarithm Power Property” states that if a logarithm has an exponent, we can take it out in front of the logarithm.