Problem 18 - Precalculus: Functions and Graphs

Question

Simplify. $$\frac{\left(3 y^{3}\right)\left(2 y^{2}\right)^{2}}{\left(y^{4}\right)^{3}} \cdot\left(y^{3}\right)^{0}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{12}{y^5}\).

Step 1: Simplify the Exponentiation (Numerator)

Start by simplifying the numerator. The expression \((2y^2)^2\) can be simplified using the power of a power rule: \((a^m)^n = a^{m \cdot n}\). This means \((2y^2)^2 = (2^1)^2(y^2)^2 = 4y^4\).

Step 2: Simplify the Exponentiation (Denominator)

Now, simplify the expression in the denominator \((y^4)^3\). Apply the power of a power rule: \((y^4)^3 = y^{4 \cdot 3} = y^{12}\). Thus, the denominator simplifies to \(y^{12}\).

Step 3: Simplify the Exponential Term

Consider the term \((y^3)^0\). Any non-zero base raised to the power of zero is 1, so \((y^3)^0 = 1\).

Step 4: Combine and Simplify the Expression

After simplifying, substitute back into the equation:\[\frac{(3y^3)(4y^4)}{y^{12}} \cdot 1\].Multiply the terms in the numerator:\(3 \cdot 4 = 12\) and \(y^3 \cdot y^4 = y^{3+4} = y^7\),so the numerator becomes \(12y^7\).

Step 5: Divide the Expression

Now divide the simplified numerator by the simplified denominator:\[\frac{12y^7}{y^{12}}\].Using the division property of exponents, \(y^m \div y^n = y^{m-n}\), we get \(y^{7-12} = y^{-5}\).The expression simplifies to \(12y^{-5}\).

Step 6: Convert Negative Exponent to Positive

Convert the negative exponent to a positive exponent by writing it as a reciprocal: \(y^{-5} = \frac{1}{y^5}\).Thus, the expression becomes \(\frac{12}{y^5}\).

Key Concepts

Exponentiation

Exponentiation is a fundamental concept in algebra that deals with raising numbers to the power of another number. It's a sort of shorthand for repeated multiplication. For instance, if you see something like \(y^3\), it means you multiply \(y\) by itself three times: \(y \times y \times y\).
This concept plays a crucial role throughout our exercise. When a base is raised to an exponent, the base is multiplied by itself as many times as the exponent indicates.
In the expression \((2y^2)^2\), each term is raised to the power of two, showcasing the power of a power rule.

• The number in front of \(y^2\) is \(2^1\), which means it too is squared, giving us \(4\).
• For \(y^2\), you multiply \(y\) by itself twice and then square those results, resulting in \(y^4\).

Power Rule

The power rule is an essential tool in exponentiation. It is summarized with the formula \((a^m)^n = a^{m \cdot n}\). This rule is particularly useful when dealing with exponents raised to another power.
In our example, both the numerator and denominator include situations where this rule applies:

• For \((2y^2)^2\), the power rule simplifies it to \(4y^4\) by calculating \(2^{2} = 4\) and \((y^2)^2 = y^4\).
• In the denominator \((y^4)^3\), this rule simplifies the expression to \(y^{12}\) by multiplying the exponents \(4 \times 3\).

The power rule allows for significant simplification of complicated expressions by reducing multiple exponentials into single exponents.

Negative Exponents

Negative exponents represent reciprocals. They allow expressions to be rewritten in ways that are sometimes more convenient for calculation. When \(y\) is raised to a negative exponent, like \(y^{-5}\), it translates to \(\frac{1}{y^5}\).
The expression we simplified, \(12y^{-5}\), demonstrates this concept. Changing it into \(\frac{12}{y^5}\) makes it easier to interpret and further utilize. Understanding negative exponents is critical because it often allows expressions to be further simplified and makes them easier to handle in mathematical equations.

• They are useful in solving equations and making expressions more manageable.
• Converting negative exponents into positive through reciprocals is a crucial step in simplification.

Division of Exponents

The division of exponents follows a straightforward rule: when you divide like bases, you subtract the exponents. Formally, this is expressed as \(y^m \div y^n = y^{m-n}\).
In our exercise, this rule simplifies \(\frac{y^7}{y^{12}}\) to \(y^{-5}\). You subtract the smaller exponent from the larger when the base remains the same, indicating how many times \(y\) is still left being multiplied in the numerator.
Mastering this division property is essential when working with polynomials and algebraic expressions involving exponents. By understanding this concept, complex problems involving division of powers or exponents become easier to navigate. It allows one to simplify expressions efficiently through basic operations on exponents.