Problem 40 - Algebra for College Students

Question

Raise each monomial to the indicated power. $$\left(-3 x y^{4}\right)^{3}$$

Step-by-Step Solution

Answer

The expression raised to the power is \(-27x^3y^{12}\).

Step 1: Identify the components of the monomial

The monomial inside the parentheses is \(-3xy^4\).The constituents of this monomial are:- A coefficient: \(-3\)- A single variable: \(x\)- Another variable raised to a power: \(y^4\).We need to find \((-3xy^4)^3\).

Step 2: Apply the power to the coefficient

Use the rule \((a^m)^n = a^{m \times n}\) to the coefficient \(-3\).Calculate \((-3)^3\).yielding \((-3)^3 = -3 \times -3 \times -3 = -27\).

Step 3: Apply the power to the variable x

Apply the power of 3 to the variable \(x\).Using the rule \((x^m)^n = x^{m \times n}\), we find \((x)^3\),which simplifies to \(x^3\).

Step 4: Apply the power to y raised to another power

Apply the power of 3 to \(y^4\).Use the same power rule: \((y^m)^n = y^{m \times n}\).This gives us \((y^4)^3 = y^{4 \times 3} = y^{12}\).

Step 5: Combine the results

Combine the results from Steps 2, 3, and 4:The expression becomes:\(-27 \cdot x^3 \cdot y^{12}\).So, \((-3xy^4)^3 = -27x^3y^{12}\).

Key Concepts

Coefficient

The coefficient is a numerical or constant factor in a term of an algebraic expression. It multiplies the variable part of the term. For example, in the expression \(-3xy^4\), \(-3\) is the coefficient.
Co-efficients determine the magnitude or scale of the term. They can be positive or negative integers, fractions, or decimals.

In our given problem, when the monomial \((-3xy^4)^3\) is raised to the third power, you also need to raise the coefficient \(-3\) to this power. This can be done by multiplying \(-3\) by itself three times:

• Calculate \((-3)^3 = -3 \times -3 \times -3\)
• The result is \(-27\), making \(-27\) the new coefficient

It's crucial to handle the coefficient correctly, especially when it involves negative numbers, as this can affect the sign of the final expression.

Variables with Exponents

Variables in algebraic expressions can be raised to a power using exponents. This implies repeated multiplication of that variable. Take \(y^4\) for instance. Here, \(4\) is the exponent indicating \(y\) is multiplied by itself four times: \(y \times y \times y \times y\).
In our problem, the expression \((-3xy^4)^3\) includes the variable \(y\) raised to an exponent of 4.

When raising the expression to the third power further, you need to raise \(y^4\) to the third power using rules of exponents. This involves:

• Calculate \((y^4)^3\)
• Multiply the exponents together: \(4 \times 3\)
• Resulting in \(y^{12}\)

It's essential to master handling variables with exponents as it helps in dealing with complex algebraic expressions more efficiently.

Power Rule for Exponents

The power rule for exponents helps in simplifying expressions involving powers of powers. The rule states that when you raise a power to another power, you multiply the exponents. In mathematical terms, if you have any base \(a\), \( \left(a^m\right)^n = a^{m \times n} \).
This concept is key to solving problems like the one we have: \((-3xy^4)^3\).
Applying this rule:

• To the coefficient: Raise the numeric coefficient \(-3\) to the third power: \((-3)^3 = -27\).
• To the variable \(x\): Apply \(x^1\) raised to the third power: \(x^{1 \times 3} = x^3\).
• To the power of \(y\):\((y^4)^3 = y^{4 \times 3} = y^{12}\).

The power rule is extremely helpful in simplifying expressions and should be a solid part of any student's algebra toolkit.