Problem 36 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ m^{10} m^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $$ m^{10} \cdot m^{2} $$ Answer: $$ m^{12} $$

Step 1: Identity the base and exponents

Here, the base "m" is the same for both terms, and their respective exponents are 10 and 2.

Step 2: Apply the product rule of exponents

Using the product rule, multiply the terms with the same base by adding the exponents: $$ m^{10} \cdot m^{2} = m^{10+2} $$

Step 3: Simplify the expression

Sum the exponents and substitute it back to the expression: $$ m^{10+2} = m^{12} $$ Now, we've simplified the expression using the product rule, and the final result is: $$ m^{12} $$

Key Concepts

Exponentiation

Exponentiation is a form of math that allows us to express repeated multiplication concisely. It involves a base and an exponent. When you see a number or variable raised to a power, like in \( m^{10} \), the base in this case is 'm' and the exponent is '10'. The exponent tells you how many times to multiply the base by itself. So \( m^{10} \) means 'm multiplied by itself 10 times'. This operation is fundamental in algebra and is used to simplify many types of expressions.

Understanding exponentiation is crucial when working with algebraic operations. It’s not just about knowing the process; it’s about comprehending the reason behind it. When you get why and how it works, simplifying expressions with exponents becomes a much easier task.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like 'm'), and arithmetic operations like addition, subtraction, multiplication, and division. They might also include exponents as part of their structure. These expressions can represent anything from simple calculations to complex relationships between variables.

When you’re simplifying these expressions, you’re essentially streamlining them without changing their value. This is like making a complicated recipe easier by combining similar steps; the outcome (the delicious meal, or in this case, the solution to the expression) remains unchanged. The goal is to make the expression as neat and as manageable as possible, whether it’s for clarity, ease of use, or to solve an equation.

Simplifying Expressions

Simplifying expressions is a bit like tidying up your room. Just as you'd group similar items together, in math, we combine like terms to tidy up expressions. With exponents, the product rule is one of the tools we use to clean up. It tells us that when we multiply terms with the same base, we add the exponents. It's important to remember that this only works when the bases are the same.

In the given problem, we used the product rule to combine \( m^{10} \) and \( m^{2} \) into \( m^{12} \). This is the mathematical equivalent of saying instead of 'writing m as a factor 10 times and then 2 more times', we can simplify it by writing 'm as a factor 12 times'.

Base and Exponent

Base and exponent go together like bread and butter in the world of mathematics. The base is the number or variable that is being multiplied by itself, and the exponent indicates how many times that multiplication occurs. In the expression \( m^{10} \), 'm' is the base and '10' is the exponent.

The rules of exponents, such as the product rule, quotient rule, power of a power rule, and others, only apply when the bases are the same. That's because these rules are based on how multiplication and division fundamentally work. When the bases match, we can use these properties to combine and simplify expressions in a way that is consistent with the overarching principles of algebra.