Simplify (2x^0y)(-3xy^5)

The given problem is a request to simplify a mathematical expression that involves algebraic terms. The terms in question are (2x^0y) and (-3xy^5). The operation to be performed between these two terms is multiplication. The problem is asking to combine these terms through multiplication and apply the relevant mathematical properties to simplify the resulting expression to its simplest form. The problem includes the use of the exponent zero, which has a specific property that affects the simplification process.

$(2 x^{0} y) (- 3 x y^{5})$

Answer

Expert–verified

Solution:

Step 1:

Combine the like bases by adding their exponents.

Step 1.1:

Rearrange the terms to facilitate multiplication: $2 (x \cdot x^{0}) y (- 3 y^{5})$ .

Step 1.2:

Multiply the base $x$ with $x^{0}$ .

Step 1.2.1:

Consider $x$ as $x^{1}$ : $2 (x^{1} \cdot x^{0}) y (- 3 y^{5})$ .

Step 1.2.2:

Apply the exponent rule $a^{m} \cdot a^{n} = a^{m + n}$ : $2 x^{1 + 0} y (- 3 y^{5})$ .

Step 1.3:

Calculate the sum of the exponents: $2 x^{1} y (- 3 y^{5})$ .

Step 2:

Simplify the expression to $2 x y (- 3 y^{5})$ .

Step 3:

Combine the like bases $y$ and $y^{5}$ by adding their exponents.

Step 3.1:

Reposition $y^{5}$ : $2 x (y^{5} \cdot y) (- 3)$ .

Step 3.2:

Multiply the base $y^{5}$ with $y$ .

Step 3.2.1:

Express $y$ as $y^{1}$ : $2 x (y^{5} \cdot y^{1}) (- 3)$ .

Step 3.2.2:

Utilize the exponent rule $a^{m} \cdot a^{n} = a^{m + n}$ : $2 x y^{5 + 1} (- 3)$ .

Step 3.3:

Sum up the exponents: $2 x y^{6} (- 3)$ .

Step 4:

Multiply the constants $- 3$ and $2$ : $- 6 x y^{6}$ .

Knowledge Notes:

The problem involves simplifying an algebraic expression with exponents. The key knowledge points and rules used in the solution include:

Exponent Rules: The power rule states that when multiplying two expressions with the same base, you add the exponents: $a^{m} \cdot a^{n} = a^{m + n}$ .
Zero Exponent Rule: Any base raised to the zero power is equal to one: $a^{0} = 1$ .
Multiplication of Constants: When multiplying constants, simply multiply their values together.
Combining Like Terms: When terms have the same variable part, they can be combined by performing the operation indicated (in this case, multiplication).
Simplification: The process of reducing an expression to its simplest form by performing all possible operations and combining like terms.

In this problem, we first apply the zero exponent rule to simplify $x^{0}$ to 1, then use the power rule to combine the exponents of $x$ and $y$ terms. Finally, we multiply the constants to reach the simplified expression.