Find the Exact Value 3*3^3
Brief Explanation:
The question asks to calculate the exact value of the mathematical expression which consists of multiplication and exponentiation. The expression given is "3 multiplied by 3 to the power of 3". You are expected to perform the arithmetic operations in the correct order following the standard rules for order of operations (PEMDAS/BODMAS), which means you should handle the exponentiation before the multiplication. The final response should be a single numerical value representing the result of these operations.
$3 \cdot 3^{3}$
Answer
Solution:
Step:1
Combine the multiplication of $3$ by $3^{3}$ using the properties of exponents.
Step:1.1
Consider the multiplication of $3$ (which is $3^{1}$) with $3^{3}$.
Step:1.1.1
Express $3$ as $3$ raised to the first power: $3^{1} \cdot 3^{3}$.
Step:1.1.2
Apply the rule for multiplying powers with the same base: $a^{m} \cdot a^{n} = a^{m + n}$. Thus, we get $3^{1 + 3}$.
Step:1.2
Perform the addition of the exponents: $3^{4}$.
Step:2
Calculate the value of $3$ raised to the fourth power: $3^{4} = 81$.
Knowledge Notes:
To solve the given problem, we need to understand the properties of exponents, specifically the power rule for multiplication. The power rule states that when multiplying two powers that have the same base, you can add the exponents. In mathematical terms, this is expressed as:
$$ a^{m} \cdot a^{n} = a^{m + n} $$
where $a$ is the base and $m$ and $n$ are the exponents.
In the given problem, we are asked to find the exact value of $3 \cdot 3^{3}$. To do this, we first recognize that the number $3$ can be written as $3^{1}$, which allows us to use the power rule. By adding the exponents, we get $3^{1+3}$, which simplifies to $3^{4}$. Finally, calculating the value of $3^{4}$ gives us $81$.
It's important to note that this rule only applies when the base numbers are the same. When the bases are different, the exponents cannot be simply added.
