Problem 80 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(3 y^{4}\right)(-5 y) $$

Step-by-Step Solution

Answer

The simplified expression is \(-15y^5\).

Step 1: Recognize the Properties of Exponents and Constants

The given expression is \((3y^4)(-5y)\). Notice that multiplication of terms involves multiplying their coefficients and adding the exponents of like bases.

Step 2: Multiply the Coefficients

Multiply the coefficients, which are the numbers in front of the variables. Here, we have \(3\) and \(-5\). So, \(3 \times (-5) = -15\).

Step 3: Apply the Law of Exponents

Identify the like bases and apply the exponent law. The common base is \(y\) with exponents \(4\) and \(1\) (since \(y = y^1\)). Add the exponents: \(4 + 1 = 5\).

Step 4: Simplify the Product

Combine the results from Steps 2 and 3. The simplified expression is \(-15y^5\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is key to simplifying algebraic expressions. Exponents reflect how many times a base number is multiplied by itself. When we have a product of powers with the same base, the exponents can be added. This is because multiplication can be thought of as repeated addition:

• For instance, \[ y^a \times y^b = y^{a+b} \]This rule makes it easier to handle expressions involving powers.
• Also, when multiplying numbers with coefficients, the coefficients are multiplied separately from the bases.

By understanding these properties, simplifying complex expressions becomes more manageable and logical.

Coefficients

Coefficients are the numerical parts of the terms in an algebraic expression. They are the numbers that multiply a variable. In the expression \( (3y^4)(-5y) \), 3 and -5 are the coefficients.

To simplify the expression, you multiply these coefficients directly. This is because they behave like normal numbers, without any exponents involved. So, \( 3 \times -5 = -15 \).

Remember, the multiplication of coefficients is separate from applying exponent rules, as those only apply to the variables. Managing coefficients effectively helps simplify the rest of the expression correctly.

Like Bases

Like bases refer to terms that share the same base in their exponential form. In the given expression \( (3y^4)(-5y) \), the variable \( y \) is the like base.

Having like bases is a crucial element when applying exponent rules. Only terms sharing the same base can have their exponents added together. This simplifies the expression while maintaining the base constant.

• For example, in the expression, \( y^4 \) and \( y^1 \) are both terms with the base \( y \). By using like bases, we add their exponents: \( 4+1=5 \), resulting in \( y^5 \).

Grasping this concept is crucial for correctly simplifying expressions with exponents.

Law of Exponents

The law of exponents provides rules for how to perform operations involving exponents. One of the most useful laws is the product of powers rule, which allows us to add the exponents when multiplying like bases:

• \[ x^a \times x^b = x^{a+b} \]

In the given expression \( (3y^4)(-5y) \), we can apply this law to the variable \( y \). Notice that \( y \) appears with an implied exponent of 1 in the second term.

Applying the law, \( y^4 \times y^1 = y^{4+1} = y^5 \).By using this exponent law, we simplify expressions efficiently without altering their meaning.