Simplify (5x^2y^5z^7)^0

The problem asks for the simplification of an algebraic expression that is raised to the power of zero. The expression given includes variables $x$ , $y$ , and $z$ , each raised to some power, and the entire expression is surrounded by parentheses and then raised to the zeroth power. The question is essentially about the application of the exponent rule that defines any nonzero number raised to the power of zero.

${((5 x^{2} y^{5} z^{7}))}^{0}$

Answer

Expert–verified

Solution:

Step 1:

Employ the exponent distribution rule $(a b)^{n} = a^{n} b^{n}$ to expand the zero exponent across all factors.

Step 1.1:

Separate the exponent of zero for the term $5 x^{2} y^{5}$ and $z^{7}$ as $(5 x^{2} y^{5})^{0} (z^{7})^{0}$ .

Step 1.2:

Further distribute the zero exponent to $5 x^{2}$ and $y^{5}$ as $(5 x^{2})^{0} (y^{5})^{0} (z^{7})^{0}$ .

Step 1.3:

Distribute the zero exponent to the coefficient and variable $5$ and $x^{2}$ as $5^{0} (x^{2})^{0} (y^{5})^{0} (z^{7})^{0}$ .

Step 2:

Recognize that any number raised to the power of zero equals one: $1 (x^{2})^{0} (y^{5})^{0} (z^{7})^{0}$ .

Step 3:

Multiply $1$ by $(x^{2})^{0}$ to maintain the expression: $(x^{2})^{0} (y^{5})^{0} (z^{7})^{0}$ .

Step 4:

Simplify the expression $(x^{2})^{0}$ by applying the power of a power rule.

Step 4.1:

Utilize the rule $(a^{m})^{n} = a^{m \cdot n}$ to simplify $x^{2 \cdot 0} (y^{5})^{0} (z^{7})^{0}$ .

Step 4.2:

Calculate $2 \cdot 0$ to find $x^{0} (y^{5})^{0} (z^{7})^{0}$ .

Step 5:

Again, apply the concept that any number to the zero power is one: $1 (y^{5})^{0} (z^{7})^{0}$ .

Step 6:

Multiply $1$ by $(y^{5})^{0}$ to keep the expression consistent: $(y^{5})^{0} (z^{7})^{0}$ .

Step 7:

Simplify the expression $(y^{5})^{0}$ by applying the power of a power rule.

Step 7.1:

Use the rule $(a^{m})^{n} = a^{m \cdot n}$ to simplify $y^{5 \cdot 0} (z^{7})^{0}$ .

Step 7.2:

Perform the multiplication $5 \cdot 0$ to get $y^{0} (z^{7})^{0}$ .

Step 8:

Recognize that any term raised to the zero power equals one: $1 (z^{7})^{0}$ .

Step 9:

Multiply $1$ by $(z^{7})^{0}$ to maintain the expression: $(z^{7})^{0}$ .

Step 10:

Simplify the expression $(z^{7})^{0}$ by applying the power of a power rule.

Step 10.1:

Apply the rule $(a^{m})^{n} = a^{m \cdot n}$ to simplify $z^{7 \cdot 0}$ .

Step 10.2:

Calculate $7 \cdot 0$ to find $z^{0}$ .

Step 11:

Conclude that any number raised to the power of zero is one: $1$ .

Knowledge Notes:

The problem-solving process involves simplifying an expression where a term is raised to the power of zero. The key knowledge points covered in this process include:

Exponent Distribution Rule: This rule states that when you have a product of terms raised to an exponent, you can distribute the exponent to each term individually. Mathematically, it is expressed as $(a b)^{n} = a^{n} b^{n}$ .
Zero Exponent Rule: Any term raised to the power of zero is equal to one. This is a fundamental rule in algebra and is expressed as $a^{0} = 1$ , where $a$ can be any real number, variable, or algebraic expression.
Power of a Power Rule: This rule is used when you have an exponent raised to another exponent. The rule states that you multiply the exponents together. It is mathematically expressed as $(a^{m})^{n} = a^{m \cdot n}$ .
Multiplication by One: Multiplying any number or expression by one does not change its value. This property is used to maintain the structure of the expression while simplifying.
Simplification: The process of reducing an expression to its simplest form by applying the above rules and performing any necessary arithmetic operations.

The solution to the given problem $(5 x^{2} y^{5} z^{7})^{0}$ involves applying these rules to conclude that the entire expression simplifies to $1$ .