Simplify (5x^4y^4z^0)^0
The question asks you to apply the rules of exponents to simplify the given mathematical expression. Specifically, the expression is a power raised to the power of zero, and you are expected to simplify it according to the properties of exponents. The expression involves variables (x and y) with exponents and one variable (z) raised to the power of zero, which is further raised to the power of zero as a whole.
$\left(\left(\right. 5 x^{4} y^{4} z^{0} \left.\right)\right)^{0}$
Answer
Solution:
Step 1
Recognize that any expression raised to the power of $0$ is equal to $1$. Thus, $(5x^4y^4z^0)^0$ simplifies to $1$.
Step 2
Since $z^0$ is equal to $1$, the expression becomes $(5x^4y^4 \cdot 1)^0$.
Step 3
The expression simplifies further to $(5x^4y^4)^0$ by multiplying $5$ by $1$.
Step 4
Invoke the power of a product rule, which states that $(ab)^n = a^n \cdot b^n$.
Step 4.1
Separate the terms under the exponent of $0$: $(5x^4)^0 \cdot (y^4)^0$.
Step 4.2
Break down the expression further into $5^0 \cdot (x^4)^0 \cdot (y^4)^0$.
Step 5
Apply the rule that any number raised to the power of $0$ is $1$: $1 \cdot (x^4)^0 \cdot (y^4)^0$.
Step 6
Simplify $(x^4)^0$ by multiplying the exponents.
Step 6.1
Use the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$: $x^{4 \cdot 0} \cdot (y^4)^0$.
Step 6.2
Calculate $4 \cdot 0$ to get $x^0 \cdot (y^4)^0$.
Step 7
Recognize again that any number raised to the power of $0$ is $1$: $1 \cdot (y^4)^0$.
Step 8
Simplify $(y^4)^0$ by multiplying the exponents.
Step 8.1
Use the power of a power rule again: $y^{4 \cdot 0}$.
Step 8.2
Calculate $4 \cdot 0$ to get $y^0$.
Step 9
Conclude that $y^0$ is also $1$, and thus the entire original expression simplifies to $1$.
Knowledge Notes:
Zero Exponent Rule: Any base (except zero) raised to the power of zero is equal to one, i.e., $a^0 = 1$ for $a \neq 0$.
Product Rule of Exponents: When multiplying two powers that have the same base, you can add the exponents, i.e., $a^m \cdot a^n = a^{m+n}$.
Power of a Product Rule: When raising a product to a power, you can apply the exponent to each factor, i.e., $(ab)^n = a^n \cdot b^n$.
Power of a Power Rule: When raising a power to another power, you multiply the exponents, i.e., $(a^m)^n = a^{m \cdot n}$.
Simplification Process: The process of simplifying an expression involves applying various arithmetic rules and properties to rewrite the expression in a simpler or more convenient form.
Multiplication by One: Multiplying any number by one leaves the original number unchanged, i.e., $a \cdot 1 = a$.
Mathematical Notation: Mathematical expressions are often written using LaTeX syntax to clearly represent numbers, variables, operators, and their respective hierarchies in calculations.
