Question

Multiply. $$\left(y^{b}\right)\left(y^{3}\right)$$

Short answer

\(y^{b+3}\)

Step-by-step solution

Recall the Exponent Rule

When multiplying two expressions with the same base, you can add the exponents according to the rule, \(a^m \cdot a^n = a^{m+n}\). Our base in both expressions is \(y\).

Add the Exponents

Since the base is the same, we add the exponents \(b\) and \(3\) together to find the new exponent: \(y^b \cdot y^3 = y^{b+3}\).

Key concepts

Exponent Rules

Understanding exponent rules is crucial when working with algebraic expressions, particularly when we involve powers of the same base. The rule applied in the exercise is known as the Product of Powers rule which tells us that when we multiply two expressions with the same base, we simply add their exponents. For instance, the expression \(y^b\) multiplied by \(y^3\) is executed by taking the base \(y\) and adding the exponents \(b\) and \(3\) to get \(y^{b+3}\).

This rule makes calculations simpler and faster, as we don't need to multiply the base by itself repeatedly for each exponent. Another key rule is the Power of a Power rule, suggesting that when raising an exponent to another exponent, like \( (y^b)^3 \), you multiply the exponents to get \(y^{b\cdot3}\). These exponent rules are foundational for working with more complex algebraic and exponential functions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain ordinary numbers, variables (like \(y\) or \(x\)), and exponents (such as \(b\) and \(3\) in our exercise). The beauty of algebraic expressions is that they represent general values and operations that can be applied to a wide range of specific cases. When working with expressions, it's important to remember that operations like addition, subtraction, multiplication, and division follow specific rules that ensure we can simplify or manipulate these expressions in a consistent way.

For example, simplifying the product of \(y^b\) and \(y^3\) relies on understanding how to combine like terms - terms with the same variable raised to a power. Here, since we are dealing with the same base, the exponents are added, reflecting the properties of exponents. This is how we translate the multiplication of powers with like bases into a single, simplified expression.

Exponential Functions

Exponential functions represent real-world phenomena such as growth and decay, characterized by a constant proportional rate of increase or decrease. They are a type of algebraic expression, usually written as \(f(x) = ab^x\), where \(b\) is the base and \(x\) is the exponent. The base \(b\) is a constant, and \(b > 0\) and \(b eq 1\).

In an exponential function, as \(x\) increases or decreases, the output of the function changes at an accelerating rate, rather than a constant rate. This behavior makes exponential functions particularly useful in modeling situations where change compounds over time, like populations, investments, and radioactive decay. When \(b > 1\), the function represents exponential growth, and when \(0 < b < 1\), it represents exponential decay. The operations that we perform on exponents, such as in the product of \(y^b\) and \(y^3\), play a significant role in shaping the behavior and graph of these functions.