Question

Answers are given at the end of these exercises. True or False If \(a\) is a real number, and \(m\) and \(n\) are integers. then \(a^{m} \cdot a^{n}=a^{m+n}\)

Short answer

True

Step-by-step solution

Understand the Given Statement

The statement given is: If \(a\) is a real number, and \(m\) and \(n\) are integers, then \(a^{m} \cdot a^{n}=a^{m+n}\). We need to determine if this is true or false.

Recall the Properties of Exponents

Recall the exponent rule: For any real number \(a\) and integers \(m\) and \(n\), the property \(a^m \cdot a^n = a^{m+n}\) holds. This is because multiplying with the same base involves adding the exponents.

Apply the Rule

Given that \(a\) is a real number and both \(m\) and \(n\) are integers, applying the exponent rule will indeed give \(a^{m} \cdot a^{n}=a^{m+n}\).

Conclusion

Since the exponent rule is applicable here and holds true, the given statement is indeed True.

Key concepts

Real Numbers

Real numbers are a fundamental concept in mathematics. They include all the numbers you can think of on a number line. This includes:

• Rational numbers: numbers that can be expressed as fractions, such as \(\frac{1}{2}\) or 2.5.
• Irrational numbers: numbers that cannot be expressed as fractions, such as π or the square root of 2.
• Integers: whole numbers that can be positive, negative, or zero.

Real numbers are significant because they form a continuous, unbroken line. This continuous nature allows us to perform a wide range of operations, including addition, subtraction, multiplication, and division (except by zero). Understanding real numbers is crucial for mastering advanced mathematical concepts, including those related to exponents.

Integers

Integers are a subset of real numbers and include all whole numbers, both positive and negative, including zero. In symbolic form, the set of integers is usually represented as \(\textbf{Z}\).

Examples of integers include:

• 3 (positive integer)
• -7 (negative integer)
• 0 (neutral integer)

.Integers are particularly important in exponent rules. When using integers as exponents, we can simplify expressions using specific properties. For instance, in the given exercise, we consider the exponents \(m\) and \(n\) as integers. This enables us to apply the multiplication rule for exponents correctly.

Multiplication of Exponents

One of the core properties of exponents is how they behave during multiplication. For any real number \(a\) and integers \(m\) and \(n\), the rule \(a^m \cdot a^n = a^{m+n}\) holds. This is known as the product of powers property.

This property works because when you multiply two numbers with the same base, you add their exponents. Here's how it works step by step:

• Start with the expression \(a^m \cdot a^n\).
• Recognize that the bases \(a\) are the same.
• Add the exponents \(m\) and \(n\) together.
• Write the resulting expression as \(\text{a}^{m+n}\).

Here's a concrete example:

If \(a = 2\), \(m = 3\), and \(n = 4\), then \(2^3 \cdot 2^4 \ = 2^{3+4} = 2^7 \). Practicing these steps will help you understand why the given exercise's statement is indeed true.