Problem 9 - Intermediate Algebra

Question

Fill in the blanks. a. An exponent indicates repeated _____. b. Subtraction is the same as adding the _____ of the number being subtracted.

Step-by-Step Solution

Answer

a. multiplication b. opposite

Step 1: Understanding Exponents

An exponent is a number that indicates how many times to multiply a base number by itself.For example, in the expression \(5^3\), the exponent is 3, indicating that 5 is multiplied by itself 3 times. Hence, an exponent indicates repeated multiplication.

Step 2: Understanding Subtraction as Addition of Opposites

Subtraction can be thought of as adding a negative number. This means that instead of subtracting a number, we add its opposite. For example, 5 - 3 can be thought of as 5 + (-3), where -3 is the opposite of 3. Therefore, subtraction is equivalent to adding the opposite.

Key Concepts

Exponents: The Power of Repeated Multiplication

Exponents allow us to express repeated multiplication in a compact form. When we have a base number raised to an exponent, it tells us how many times to multiply the base by itself. For example, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent, meaning \(a\) is multiplied \(n\) times.

• “Squaring” a number is when the exponent is 2. For example, \(4^2\) is \(4 \times 4\) which equals 16.
• “Cubing” a number means using an exponent of 3, such as \(3^3\) which translates to \(3 \times 3 \times 3\) equaling 27.

Repetition here shows how exponents simplify complex multiplication. This helps in solving larger problems efficiently.

Subtraction and the Addition of Opposites

Subtraction might feel straightforward but it has underlying principles that make it even more intuitive. It can be reframed as the addition of an opposite number. This idea is especially useful when dealing with both positive and negative numbers. For example, subtracting \(3\) from \(10\) is the same as adding \(-3\) to \(10\):

• 10 - 3 is equivalent to 10 + (-3)
• Both calculations will yield the same result: 7

By viewing subtraction as addition of opposites, complex arithmetic becomes simpler and it broadens your understanding when dealing with algebraic expressions.

Understanding Repeated Multiplication with Exponents

Repeated multiplication is the foundation on which exponents are built. Instead of writing expressions with multiple multiplication signs, you can use exponents to simplify them. For example:

• \(2 \times 2 \times 2 \times 2\) can be efficiently written as \(2^4\).
• This expression, \(2^4\), means multiplying 2 by itself 4 times, resulting in 16.

Another common example is multiplying one number several times, like \(5 \times 5 \times 5\) can be simplified to \(5^3\). Repeated multiplication and exponents are closely linked, making calculations more manageable especially with larger numbers.