Problem 39 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$\left(6 y^{2}\right)\left(2 y^{3}\right)^{2}$$

Step-by-Step Solution

Answer

The short version of the answer is: \(24y^{8}\).

Step 1: Apply the power of a power rule

Using the power of a power rule, which states that \((a^m)^n = a^{m \times n}\), rewrite the expression: \[ \left(6 y^{2}\right)\left(2 y^{3}\right)^{2} \] becomes \[ \left(6 y^{2}\right)\left(2^2 \cdot y^{3 \cdot 2}\right) \]

Step 2: Simplify the exponents and constants

Now, simplify the numerical constants and the exponents: \[ \left(6 y^{2}\right)\left(4 \cdot y^{6}\right) \]

Step 3: Apply the product of powers rule

Apply the product of powers rule, which states that \(a^m \cdot a^n = a^{m+n}\), to multiply the \(y\) terms: \[ 6y^2 \cdot 4y^6 \] There's no need to apply this rule to the numerical constants, simply multiply them: \[ 24y^2y^6 \]

Step 4: Final simplification

By applying the product of powers rule again to simplify the y variables: \[ 24y^{2+6} \] The final simplified expression is: \[ 24y^{8} \]

Key Concepts

Power of a Power Rule

The power of a power rule is one of the fundamental rules in algebra that helps in simplifying expressions where exponents are involved. It states that when you have an expression of the form \((a^m)^n\), you can simplify it by multiplying the exponents: \(a^{m imes n}\). This rule is essential when dealing with expressions that have been raised to another power. It helps in condensing the expression, making it easier to manage and solve.

• For example, \((x^2)^3\) becomes \(x^{2\times3} = x^6\).
• In the given exercise, \((2y^3)^2\) was simplified using this rule as \(2^2y^{3\times2}\), resulting in \(4y^6\).

Understanding this rule makes simplifying complex expressions more straightforward and less intimidating. By applying this rule, we reduce the steps needed to solve an equation, thus minimizing errors.

Product of Powers Rule

The product of powers rule is another key concept in algebra that involves combining terms with the same base. This rule states that \(a^m \cdot a^n = a^{m+n}\). It allows you to add the exponents together when you are multiplying terms with the same base. This can simplify expressions considerably when handling variables raised to powers.

• For instance, \(x^4 \cdot x^3\) simplifies to \(x^{4+3} = x^7\).
• In our example, after simplifying the expression to \(6y^2 \cdot 4y^6\), this rule was used to get \(y^{2+6}\), resulting in \(y^8\).

By understanding the product of powers rule, students can tackle problems efficiently, combining terms with ease and enhancing their fluency in solving algebraic expressions.

Exponents

Exponents are a shorthand way of expressing repeated multiplication of a number by itself. An exponent is written as a small number, called the power, placed to the upper right of the base number. The base is the number that is multiplied, and the exponent signifies how many times to use the base in the multiplication. For example, \(5^3\) means \(5 \times 5 \times 5\).

• The expression \(y^3\) means \(y \times y \times y\).
• In the solved example, exponents were used to represent both the constants and variables efficiently.

Exponents are crucial in math because they provide a clean and efficient way to represent and calculate large numbers. Mastering the rules that govern their manipulation, like the power of a power and product of powers rules, greatly enhances mathematical skills.

Polynomials

Polynomials are expressions consisting of variables and coefficients, using operations of addition, subtraction, multiplication, and non-negative integer exponents. They are widely used throughout mathematics to model various phenomena and to solve equations. A polynomial can be as simple as \(x^2+2x+1\), or it can include many terms, depending on its degree.

• The degree of a polynomial is determined by the highest power of the variable in the expression.
• In our example, after simplification, the expression can be considered a polynomial in the variable \(y\).

Understanding polynomials and their properties is foundational in algebra. They allow us to better comprehend the structure and function of equations, giving insight into their universe of solutions.